Stability phenomena in the topology of moduli spaces

Stabilit y phenomena in the top ology of mo duli spaces Ralph L. Cohen ∗ Dept. of Mathematics Stanford Univ ersit y Stanford, CA 943 05 Octob er 30, 2018 Abstract The recent proof by Madsen and W eis s of Mumford’s con jecture on the stable cohomology of mod u li spaces of Riemann su rfaces, w as a d ramatic example of an imp ortant stability th eorem abou t the top ology of mo duli spaces. In this article we give a survey of families of classifying spaces and moduli spaces where “stabilit y phenomena” o ccur in their topologies. Such stabilit y theorems h a ve b een pro ved in man y situ ations in the history of top ology and geometry , and the pay off has often b een quite remark able. In t h is pap er we discuss classical stability theorems such as the F reudenthal susp ension theorem, Bott p eriod icit y , and Whitney’s embedding theorems. W e then discuss more mo dern examples su c h as th ose involving configuration spaces of p oints in manifolds, h olomorphic curves in complex manifol ds, gauge theoretic mo duli spaces, the stable top ology of general linear groups, and pseudoisotopies of manifolds. W e th en discuss the stabil ity theorems regarding the mo duli spaces of Riemann su rfaces: Harer’s stability theorem on t he cohomology of modu li space, and the Madsen-W ei ss theorem, which pro ves a generalization of Mumford’s conjecture. W e also describe Galatius’s recen t theorem on th e stable cohomology of automorphisms of free groups. W e end by specu lating on the existence of general conditions in whic h on e might exp ect these stability ph enomena to occur. Con ten ts 1 Classical stabili t y theorems 4 1.1 The F reudenthal suspensio n theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.2 Whitney’s E mbedding Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.3 Bott p erio dicity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 ∗ The author was partially supported b y NSF gran t DMS-0603713 1 2 Configuration spaces, p ermutations, and braids 8 2.1 Configuratio ns of points in a manifold . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.2 Symmetric gr oups and braid gr oups . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 3 Holomorphi c curv e s and gauge theory 11 3.1 Holomorphic Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 3.2 Gauge theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 3.2.1 Flat connections on Riemann surfaces . . . . . . . . . . . . . . . . . . . . . . 1 4 3.2.2 Self dual co nnections on four-ma nifolds and the Atiy ah-Jones Conjecture . . 16 4 General l inear groups, Pseudoi s otopies, and K -theo ry 18 4.1 The stable top olo gy of genera l linear groups and a lg ebraic K -theo ry . . . . . . . . . 18 4.2 Pseudoiso to pies, and W aldhausen’s alge br aic K -theory of spa c es . . . . . . . . . . . 1 8 5 The mo duli space of Riemann surfaces, mapping class groups, and the Mumford conjecture 20 5.1 Mapping class g r oups, mo duli spaces, and Thom spaces . . . . . . . . . . . . . . . . 2 0 5.2 Automorphisms o f free groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 6 Final Com men ts 29 In tro duction In the last sixty years, the notions of classifying spa ce and mo duli spa ce hav e pla yed central roles in the developmen t of top olog y and geometry . These ar e s paces that enco de the basic top ologica l or geo metric structure to b e studied, and therefo re the to po logy of these spaces naturally hav e b een a sub ject of intense int erest. Proba bly the mos t fundamen tal among them a re the mo duli spaces of Riemann surfac es of genus g , M g . In a dra matic application of alg ebraic top ological metho ds to algebraic geometr y , Madsen and W eiss recently pro ved a well known conjecture of Mumford re g arding the stable cohomolog y of moduli space [4 2]. Namely , Mumford describ ed a ring homomorphis m fro m a gr a ded p olynomial algebra ov er the rationals, to the cohomology of moduli space with rational co efficients, Q [ κ 1 , κ 2 , · · · κ i , · · · ] − → H ∗ ( M g ; Q ) , and c onjectured that it is an isomor phis m when the genus g is la rge with r esp ect to the coho mological grading. Here κ i is the Miller -Morita- Mumfor d cano nical class, and has gr ading 2 i . In [4 2] Madsen and W eiss des crib ed a homotop y theoretic mo del for the stable moduli space, M ∞ , and in so do ing , not only prov ed Mumford’s conjecture, but also gav e an implicit mo del for the stable cohomolo gy of mo duli space with any co efficients. Using this explicit mo del, Gala tius [20] calculated this stable 2 cohomolog y explicitly , when the co efficients are Z /p for p any prime, and in so doing uncovered a v a st amount of previously undetected tor sion in the stable co homology of mo duli spa ce. The Mads e n- W eiss theorem can b e viewed as one o f the most recent examples o f a stability theorem r e garding the top olog y of classifying spaces or mo duli space s . The purp os e of this pap er is to give a sur vey of these types of theorems a nd their applications to a br oad rang e of topics in top ology and g e ometry . Stabilit y theor ems are r esults reg arding fa milies o f classifying spa c es or mo duli spaces. These spaces are typically indexed by some geometr ic ally defined quantit y , such as the degr ee of a map, the rank o f a bundle, the g enus of a curve, or a c haracter istic num ber. W e r efer to this num ber as the “degr e e”. (In the case o f the mo duli spa ces of curves, this index ing degree is the genus of the curve.) W e let M d be the mo duli spac e c o rresp onding to degr ee d . Two basic q ue s tions ab out the topolo g y of these spa ces naturally o ccur, and seeing how they are addressed in a v ariety of examples is the ba s ic theme of this paper . • Stability Question 1. How do es the top ology o f the mo duli spa ces change a s the degr ee changes? Is there a “stability rang e” for their homolo gy or homo topy groups? By this w e mean a function r ( d ) which is an unbounded and nondecrea sing function of the degree d , with the prop er t y that the k th homology and/or homotopy group o f M d and M d +1 are iso morphic so long as k < r ( d ). • Stability Q uestion 2. Is there a naturally defined, more easily access ible limiting homotopy t yp e, as the degree gets large? If so, calculate this “stable homotopy type” as explicitly as po ssible. In this survey article w e discuss a v a riety of exa mples of families of class ifying spa ces and mo duli spaces wher e thes e questions hav e be en addressed. Different tec hniques have b een used to study these questions , but as we hope to point out, there a re common themes among these techniques. W e orga nize this survey in the fo llowing w ay . In s ection one, we discuss class ical stabilit y the- orems, including the F reudenthal sus p ensio n theor em, Bott’s perio dicit y theo r em, and Whitney’s embedding theo rems. In sections 2 thr ough 4 we discuss more mo dern s ta bilit y theorems, includ- ing those dealing with configura tion spaces of p oints in manifolds, holo morphic curves in co mplex manifolds, gauge theore tic mo duli spa ces, the stable topo lo gy of general linear groups, and ps e u- doisotopies of manifolds . In section 5 w e discuss the ba ckground o f Mumford’s conjecture, including the s tability theo rem for mapping class g roups of Harer. W e then discuss the Madse n- W eiss theore m in so me deta il, and als o descr ib e similar theore ms rega rding automor phism gro ups of free g roups. This includes stability theo rems of Hatcher a nd V og tmann, and the rece nt theo rem of Gala tius ab out the s table cohomolog y of automorphisms of free g r oups. W e end with a discussion in section 6 reg arding p otential resea r ch ques tio ns whose goal is to find general criter ia under whic h stability theorems hold (and do not hold). 3 1 Classical stabilit y theorems 1.1 The F reuden t hal susp ension theorem Probably the oldest e x ample o f a stability theorem in top ology a nd geo metry , proved in 1938, is the “F reudenthal sus pens ion theorem” [1 9]. Let Ω d S d be the s pa ce of s elf maps of the sphere S d = R d ∪∞ that fix the ba s epo int at infinity . By the adjoint co nstruction, there is a natural identification of homotopy gr oups, π q Ω d S d ∼ = π q + d S d . Moreov er there is a natura l “susp ension” map Σ : Ω d S d → Ω d +1 S d +1 defined as follows. Le t X b e a ny space with a fixed ba sep oint x 0 ∈ X . The susp ension of X , written Σ X is the quotient Σ X = S 1 × X/ ( ∞ × X ) ∪ ( S 1 × x 0 ) . This construction is natural, in the se nse that if one has a (basep oint preserving) map f : X → Y , then one has a n induced “susp ension map”, Σ f : Σ X → Σ Y defined by Σ f ( t, x ) = ( t, f ( x )) . There is a natural ide ntification (homeomo r phism) of Σ S d ∼ = S d +1 , with resp ect to whic h the susp ension construction defines the map Σ : Ω d S d → Ω d +1 S d +1 . The following is F reuden thal’s basic theorem: Theorem 1. The sus p ension map Σ : Ω d S d → Ω d +1 S d +1 induc es an isomorphism in homotopy gr oups Σ ∗ : π q (Ω d S d ) ∼ = − → π q (Ω d +1 S d +1 ) for q < d − 1 . It is a surje ction for q = d − 1 . In other wor ds Σ ∗ : π r S d → π r +1 S d +1 is an isomorphi sm for r ≤ 2( d − 1) , and is a surje ction for r = 2 d − 1 . Notice that this result can be viewed as answering Stability Question 1 in this setting. This theorem has the following generalization. Let X be any k - c o nnected space with a distinguished basep oint, x 0 ∈ X . That is, π r X = 0 for r ≤ k . L e t Ω d X denote the space o f contin uous maps α : S d → X that ta ke the basepo int ∞ ∈ S d to x 0 . Susp ending defines a map Ω d X → Ω d +1 Σ X . The following giv es a gener alization of the ab ov e theorem: Theorem 2. Σ ∗ : π q (Ω d X ) → π q (Ω d +1 Σ X ) is an isomorp hism for q ≤ 2 k − d , and is s u rje ctive for q = 2 k − d + 1 . In other wor ds, Σ ∗ : π j X → π j +1 (Σ X ) is an isomorp hism for j ≤ 2 k and is a surje ction for j = 2 k + 1 . 4 As mentioned, these results can b e viewed as answ ers to Stabilit y Question 1 in this co n text. T o a ddress Stability Question 2, one considers the limiting space, Q ( X ) = lim n →∞ Ω n Σ n X . The homotopy gr oups of Q ( X ), are the stable homo topy g r oups of X , π q ( Q ( X )) = lim n →∞ π q + n Σ n ( X ) = π s q ( X ) . While these stable ho mo topy gro ups ar e notor iously difficult to compute, they do hav e a significant adv an tage ov er the unstable homoto p y gr oups. Na mely , the functor X → π s ∗ ( X ) is a (r e duced) generalized homo logy theo ry , in that it satisfies the Eilenberg- Steenr o d axio ms. In particular the excision axiom holds for the stable theory , but do es not hold for unstable homoto py groups. Over the years this has allow ed for a v ariet y of p ow erful calculational techniques. An imp o rtant one, for example, is the spectr al sequence of Atiy ah a nd Hirzebruch that appr oximates π s ∗ ( X ) by the homology gro ups, H ∗ ( X ; π s ∗ ), where the coefficients, π s ∗ are the sta ble homotopy groups of spheres. 1.2 Whitney’s E m b edding Theorem The classical Whitney E mbedding Theorem can b e viewed as a stability theor em for the mo duli space of smo o th submanifolds of R ∞ of a giv en diffeomorphism t ype. Mor e specifically , let M n be a close d n -dimensional smo o th manifold. Let E mb ( M n , R N ) b e the space of smoo th embeddings e : M n ֒ → R N . This embedding space is top olog ized using the compact-op en topolo gy . Whitney’s basic embedding theo rem [62] is the fo llowing. Theorem 3. F or N ≥ 2 n , E mb ( M n , R N ) is nonempty. F or N ≥ 2( n + k ) , the sp ac e E mb ( M n , R N ) is ( k − 1) -c onne cte d. That is, the homotopy gr oups, π i ( E mb ( M n , R N )) = 0 for i ≤ k − 1 . Notice that in the ab ov e theorem for k = 1, it s ays that E mb ( M n , R 2 n +2 ) is connected; i.e. any t wo embeddings ar e isotopic. The fact that π 1 ( E mb ( M n , R 2 n +4 )) = 0 can b e interpreted to say that not only are any t wo embeddings iso topic, but any t w o isotopies can b e deformed to each other b y a one-para meter family of isotopies . T aking the limit as N → ∞ , one has that E mb ( M n , R ∞ ) is weakly contractible (i.e. all of its homotopy groups ar e zero ). Indeed it ca n easily b e shown that this space is con tractible, which can be interpreted as sa ying that not only are any tw o em beddings isotopic, but that there is a contractible family of choices of isotopies b etw een them. The diffeomorphism gr oup Diff ( M ) a cts fr e ely on the em bedding spaces, E mb ( M n , R N ). The action a lso is known to admit slices, which implies that the pro jection onto the quotient, which w e call M N ( M ) , is a fib er bundle. W e can think of M N ( M ) as the mo duli space of submanifolds of R N that a re diffeomor phic to M . As a consequence of Whitney’s theo rem, one has the answer to Stabilit y Q uestion 1 in this con text. 5 Corollary 4. The line ar inclusion R N ֒ → R N +1 induc es a “gluing map” M N ( M n ) → M N +1 ( M n ) which induc es an isomorph ism in homotopy gr oups in dimensions less than  N − 2 n 2  , and is su rje ct ive in dimension  N − 2 n 2  . Now by le tting N → ∞ , Whitney’s theorem als o s upplies an answer to Stability Questio n 2 in this s etting. Namely , since Whitney’s theorem implies that the total space of the bundle E mb ( M n , R ∞ ) → E mb ( M n , R ∞ ) / Diff ( M n ) = M ∞ ( M ) is w eakly con tractible, the mo duli space can b e taken to b e the c la ssifying space of the diffeomorphism group, M ∞ ( M ) ≃ B Diff ( M n ) . This obser v a tion can be interpreted in the following w ay . Consider the mo duli space with o ne marked p oint, M ∞ , 1 ( M ) = { ( N , x ) , where N ⊂ R ∞ is diffeomorphic to M , and x ∈ N . } The pro jection map p : M ∞ , 1 ( M ) → M ∞ ( M ) ( N , x ) → N is a fib er bundle who se fib er is M . It is r eferred to as the “ca no nical” M - bundle ov er M ∞ ( M ). The following in terpretation of M ∞ ( M ) as the c lassifying space B Diff ( M ) has be e n used in an impo rtant way by Ma dsen and W eiss in their pro of of Mumford’s conjecture on the s ta ble homology of the mo duli space of curves [4 2], as well as in the study of cob ordism categor ies [22]. Prop ositio n 5. The stable mo duli sp ac e of manifolds diffe omo rphic to M , M ∞ ( M n ) , classifies fib er bund les with fib er M n . That is, for a sp ac e X of the homotopy t yp e of a C W -c omplex, ther e is a bije ctive c orr esp ondenc e, φ : [ X , M ∞ ( M )] ∼ = − → B dl M ( X ) wher e the left hand side is the set of homotopy classes of maps, and t he right hand side is the set of isomorph ism classes of fib er bund les over X with fib er M and structur e gr oup Diff ( M ) . The c orr esp ondenc e φ assigns to a map f : X → M ∞ ( M ) the pul l-b ack of the c anonic al bun d le, f ∗ ( M ∞ , 1 ( M )) . 1.3 Bott p erio dicity In [7] R. B ott proved his famous p er io dicity theor em on the “ stable” homotopy type of Lie groups . Primarily this is a theore m a b o ut the homotopy type of the ortho g onal gro ups and unitary gr o ups 6 O ( n ) a nd U ( n ) as n gets lar g e. The s e res ults can be interpreted as stability results ab out the mo duli space of vector spaces, in the following wa y . Let Gr k ( C n ) b e the Grassmannian of k -dimensional complex subspa ces of C n . By increasing n , one can co nsider the infinite Grass mannian Gr k ( C ∞ ). In analo gy with the ab ov e discuss ion ab out embeddings of manifolds , this Grassmannian can be v ie wed as the quotient , Gr k ( C ∞ ) = M ono ( C k , C ∞ ) /U ( k ) where M ono ( C k , C ∞ ) is the space of linear monomor phisms that preserve the Hermitian inner pro duct (the “Stiefel manifold”). This spa c e is acted upo n freely by the unitary gr oup, U ( n ), and the quotient space, Gr k ( C ∞ ) c an be viewed as the “moduli space” of k - dimensional complex vector spaces. Since M ono ( C k , C ∞ ) is contractible, this space is a model for the classifying space, B U ( k ), which classifies k -dimensional complex vector bundles. (See [47] for a thoro ugh discussion.) Given a k -dimensio na l spac e V ⊂ C ∞ , then crossing with a line g ives V × C ⊂ C ∞ × C . Choo sing a fixed iso morphism C ∞ × C ∼ = C ∞ defines a “gluing” map, g k : Gr k ( C ∞ ) → Gr k +1 ( C ∞ ) . It is well known (see [47]) that this map is homotopy equiv alen t to the unit sphere bundle π k +1 : S ( γ k +1 ) → Gr k +1 ( C ∞ ) where S ( γ k +1 ) = { ( W , w ) : W ∈ Gr k +1 ( C ∞ ) , and w ∈ W with | w | = 1 . } Since the fib er of π k +1 is the sphere S k , one has the following answer to Stability Ques tion 1 in this con text: Prop ositio n 6. The gluing map g k : B U ( k ) → B U ( k + 1) induc es an isomorphism in homotopy gr oups in dimensions less than k ,and is a surje ction in dimen- sion k . In this setting, Bott’s theorem, one of the most imp ortant theorems in topo logy in the tw en tieth centu ry , can b e viewed as an answer to Stability Q uestion 2. Let B U = lim k →∞ B U ( k ) b e the (homotopy) c o limit of the gluing maps g k . Theorem 7. (Bott p erio dicity [7]) π q ( Z × B U ) ∼ =    Z if q is even 0 if q is o dd In p articular these homotopy gr oups ar e p erio dic, with p erio d 2. If B O is define d similarly (u sing r e al Gr assmannians), then π q ( Z × B O ) is p erio dic of p erio d 8, and the first eight homotop y gr oups (s tarting with dimension 0) ar e given by Z , Z / 2 , Z / 2 , 0 , Z , 0 , 0 , 0 . 7 2 Configuration spaces, p erm utatio ns, and braids 2.1 Configurations of p oin ts in a manifold Let M b e a manifold, and F k ( M ) ⊂ M k be the spa ce of k -dis tinct points in M . There is a natura l free action of the symmetric group, Σ k , and we let C k ( M ) b e the orbit space, C k ( M ) = F k ( M ) / Σ k . C k ( M ) is the mo duli space of k - p oints (or pa rticles) in M and ha s proven extremely imp ortant in a v ariet y of applications in top olog y , geometry , and physics. Generalizing res ults o f Sega l [5 2], McDuff pr ov ed results that answer Stabilit y Ques tions 1 and 2 in this context. These results can be describ ed a s follows. Let M be a smo oth, c onnected, o pe n n -dimens io nal manifold that is the int erior of a compact manifold with b ounda r y , ¯ M . Let p : T M → M b e the tangen t bundle, and let T ∞ M → M be the a sso ciated S n -bundle obtained by taking the fib erwise one p oint c o mpactification of T M . So the fib er of p : T ∞ M → M at x ∈ M , is the compactified tangen t s pa ce, T x M ∪ ∞ . Let Γ M be the spa ce of smoo th s e c tions of T ∞ M that hav e co mpa ct suppo rt. Such a section has a degree. W e write Γ( M ) = ` k ∈ Z Γ k ( M ), where Γ k ( M ) are the sections o f degre e k . It is not hard to see that the ho motopy t ype of Γ k ( M ) is indep endent of k . Theorem 8. [44] Ther e ar e gluing m aps γ k : C k ( M ) → C k +1 ( M ) and a family of maps α k : C k ( M ) → Γ k ( M ) that satisfy the fol lowing pr op erties: 1. The induc e d map in homolo gy ( α k ) ∗ : H q ( C k ( M )) → H q (Γ k ( M )) is an isomorp hism if k is sufficiently lar ge. 2. These homomorphi sms ar e c omp atible in the sense that the fol lowing diagr ams c ommute: H q ( C k ( M )) ( α k ) ∗ − − − − → H q (Γ k ( M )) γ k   y   y ∼ = H q ( C k +1 ( M )) ( α k +1 ) ∗ − − − − − → H q (Γ k +1 ( M )) wher e the right vertic al map is induc e d by a homotopy e qu ivalenc e Γ k ( M ) ≃ − → Γ k +1 ( M ) . Observe that this theor em answers b oth Stability Questions 1 and 2 for these moduli spaces . Question 1 is answered b ecause this theorem s ays that the gluing ma ps γ k : C k ( M ) → C k +1 ( M ) induce isomorphisms in homology through a r ange of dimensions. In fact it is prov ed that the maps γ k induce mono morphisms in homo logy in all dimensions. Q uestion 2 w as answered b ecaus e this theorem implies the following. Let C ( M ) b e the (ho mo topy) colimit, C ( M ) = lim k →∞ C k ( M ), where the limit is ta ken with resp ect to the gluing maps γ k . 8 Corollary 9. The maps α k induc e a map α : Z × C ( M ) → Γ( M ) which induc es an isomorphism in homolo gy. Notice that if M has a trivial tangent bundle, then Γ( M ) ∼ = C ∞ cpt ( M , S n ), the space of smo o th maps with compact supp ort. This in turn is ho mo topy equiv alen t to M ap • (( M ∪ ∞ ) , S n ), wher e M ap • denotes the space o f basepo in t preserving con tinuous maps. Moreov er, if ¯ M is a compact manifold with b oundary , having M as its interior, this space can be view ed as the space of maps of pairs, M a p (( ¯ M , ∂ ¯ M ) , ( S n , ∞ )). An impor tant sp ecial ca se of this theor em, whic h was prov ed prior to the pro of of Theorem 8 is when M = R n . One then has the following w ell known theorem a bo ut configura tions of p oints in Euclidean space [5 2] [43]. Theorem 10. Ther e ar e maps α k : C k ( R n ) → Ω n k S n with the fol lowing homolo gic al pr op erties: 1. ( α k ) ∗ : H ∗ ( C k ( R n )) → H ∗ (Ω n k S n ) is a monomorphism in al l dimensions. 2. ( α k ) ∗ : H q ( C k ( R n )) → H q (Ω n k S n ) is an isomorp hism is k is sufficiently lar ge with re sp e ct to q . 3. α : Z × C ( R n ) → Ω n S n induc es an isomorphism in homolo gy. Her e Ω n k S n is the sp ac e of ( b asep oi nt pr eserving) self maps of S n of de gr e e k . 2.2 Symmetric gr oups and braid groups Two sp ecial cases of The o rem 10 are w orth pointing o ut. Fir st we consider the case when n = ∞ . In this case the space of ordered configurations of points, F k ( R ∞ ) are contractible. T o se e this, one considers the pro jection fibrations, p k : F k ( R m ) → F k − 1 ( R m ) given by pro jecting onto the firs t k − 1 co ordinates. The fiber of this fibr ation is R m − { k − 1 } , Euclidea n s pace with ( k − 1)-p oints removed. This space ha s the ho motopy t ype of a wedge of ( k − 1) spher es of dimension m − 1, a nd therefor e its homotopy gr oups a re zero through dimension m − 2. An inductive argument (on k ), then shows that π q ( F k ( R m )) = 0 for q ≤ m − 2. W e therefor e hav e that F k ( R ∞ ) is contractible, a nd has a fre e a c tio n of the sy mmetr ic group Σ k . Thus C k ( R ∞ ) is a mo del for the cla s sifying space B Σ k . Therefore its (co)homolo gy is the (co)homology of the symmetric gro up Σ k . An alternative v iewp o int is that C k ( R ∞ ) is the mo duli space B Diff ( M ), as considered in the last section, where M is the z ero dimensional manifold con- sisting of k - points. In any case, Theorem 10, applied to the case n = ∞ giv es the following theorem known as the “Bar ratt-Priddy- Quillen theorem” [5]. Notice that it addr esses bo th Stability Ques- tions 1 and 2 in for the mo duli s pace of p o ints. This result b egan a fundamentally impor tant line of resear ch reg arding the relationship of finite group theo r y to stable homotopy theory . This line of 9 resear ch remains quite active to day , more than 3 5 years after the pr o of of the B arratt-P riddy-Quillen theorem. Theorem 11. Ther e ar e maps α k : B Σ k → Ω ∞ k S ∞ with the fol lowing homolo gic al pr op erties: 1. ( α k ) ∗ : H ∗ ( B Σ k ) → H ∗ (Ω ∞ k S ∞ ) is a monomorphism in al l dimensions. 2. ( α k ) ∗ : H q ( B Σ k ) → H q (Ω ∞ k S ∞ ) is an isomorphism if k is sufficiently lar ge with r esp e ct to q . 3. α : Z × B Σ ∞ → Ω ∞ S ∞ induc es an isomorphi sm in homolo gy. Her e Ω ∞ k S ∞ = lim n →∞ Ω n k S n . Another imp ortant special ca se of Theor em 10 is when n = 2. W e note that F 2 ( R 2 ) has the ho - motopy type of S 1 , whose homotopy groups are Z in dimension o ne , and zer o in a ll other dimensions . In other words, S 1 is an Eile nber g-MacLa ne spac e , K ( Z , 1). The fibra tion p k : F k ( R 2 ) → F k − 1 ( R 2 ) has fib er R 2 − { k − 1 } , w hich has the homo topy t yp e of a w edge of k − 1 circles, whic h is a lso a K ( π , 1). Now an e a sy inductive argument (on k ) implies that eac h F k ( R 2 ) is a K ( π , 1), a s is the quotient, C k ( R 2 ), for appr o priate groups π . In the case of C k ( R 2 ), its fundamen tal group is Artin’s braid group, β k . One can easily visualize that a one-pa rameter family of configur ations of k unorde r ed po int s in the plane can b e identified with is a braid in R 3 . So C k ( R 2 ) is the classifying space of Artin’s braid g roup β k . Mor e ov er, the natur a l inclus io n, C k ( R 2 ) ֒ → C k ( R ∞ ) is a map B β k → B Σ k , the ho motopy type of which is determined b y the homomorphism β k → Σ k that sends a bra id to the resulting per mutation of the ends o f the s tr ings. F urthermore , the covering space Σ k → F k ( R 2 ) → C k ( R 2 ) makes it appa rent that F k ( R 2 ) is the classifying s pa ce for the pure braid g r oup, P β k , which is the k ernel of β k → Σ k . The sp ecial case of Theo rem 10 in the cas e n = 2, establishes the close connections betw een Artin’s braid gro ups and self maps of S 2 : Theorem 12. Ther e ar e maps α k : B β k → Ω 2 k S 2 with the fol lowing homolo gic al pr op erties: 1. ( α k ) ∗ : H ∗ ( B β k ) → H ∗ (Ω 2 k S 2 ) is a monomorph ism in al l dimensions. 2. ( α k ) ∗ : H q ( B β k ) → H q (Ω 2 k S 2 ) is an isomorphism is k is sufficiently lar ge with r esp e ct to q . 3. α : Z × B β ∞ → Ω 2 S 2 induc es an isomorphism in homolo gy. W e end this section by recalling that one o f the applications of these configuratio n s paces is that they may b e viewed as ho mogeneous spaces in the following se ns e. Supp ose, like a bove, that M is the interior of a manifold with b oundar y ¯ M . Let Diff ( ¯ M , ∂ ) b e the group of diffeomorphisms o f ¯ M that fix the b oundar y , ∂ ¯ M po int wise. If M is oriented, w e wr ite Diff + ( ¯ M , ∂ ) to denote the s ubg roup of diffeomorphis ms that preserve the orientation. 10 Notice that Diff ( ¯ M , ∂ ) acts transitively on the configura tion spa ce C k ( M ), and the isotropy group of a fixed configuration of k points is the subgroup Diff ( ¯ M , { k } , ∂ ) that fix those k -po in ts (a s a set). This gives a homeomorphism from C k ( M ) to a homogeneous space, C k ( M ) ∼ = Diff ( ¯ M , ∂ ) / Diff ( ¯ M , { k } , ∂ ) . (1) Similarly , if M is oriented, it can b e written as the quotient, C k ( M ) ∼ = Diff + ( ¯ M , ∂ ) / Diff + ( ¯ M , { k } , ∂ ) . In the case when M = D 2 , the ope n, t wo-dimensional disk, then a famous theorem of Smale as- serts that Diff + ( ¯ D 2 , ∂ ) is c o nt ractible. Thus the quotient Diff + ( ¯ D 2 , ∂ ) / Diff + ( ¯ D 2 , { k } , ∂ ) ∼ = C k ( R 2 ) is the classifyng o f the diffeomorphism g roup, C k ( R 2 ) ≃ B Diff + ( ¯ D 2 , { k } , ∂ ). Now as w e saw ab ov e, C k ( R 2 ) is a K ( π , 1), w hich implies that the ho motopy g r oups of the diffeomorphis m gr oup Diff + ( ¯ D 2 , { k } , ∂ ) ar e zero in p ositive dimensions. This is equiv alen t to saying that the subgr oup of diffeomorphisms that are isotopic to the iden tit y is w eakly con tractible. In particular this says that the discr ete group o f isotopy classes of diffeomo rphisms Γ( ¯ D 2 , { k } , ∂ ) is the fundamental gro up of C k ( R 2 ). In general, the gr oup of isotopy classes of diffeomor phisms o f a surface is known as the mapping class group of that surface. (See Sectio n 5 for a more complete discus sion.) In particular this says that the braid gr oup can b e viewed as the mapping cla ss group, β k ∼ = Γ( ¯ D 2 , { k } , ∂ ) . Thu s Theorem 12 can b e interpreted as a stability result for the homolo g y of these mapping clas s groups. Stabilit y theorems for mapping class gr oups of p ositive genus s ur faces will b e the main sub ject of section 5 below. 3 Holomorphic curv es and gauge theory In this section we discuss mor e mo dern stability theorems that lie in the in tersection of top olo gy and a lgebraic and differential geo metry . These ar e sta bilit y theorems r egarding mo duli spaces of holomorphic maps, bundles , and Y ang-Mills co nnections. 3.1 Holomorphic Curv es The first stability theorem reg a rding mo duli spac es of holomo r phic curves was due to Segal [5 4]. Let Rat d ( CP m ) be the space of bas ed r ational maps in C P m of de g ree d . That is, Rat d ( CP m ) consists holomorphic maps α : CP 1 → CP m that take ∞ ⊂ CP 1 = C ∪ ∞ , to [1 , 1 , · · · , 1] ∈ CP n , and have degre e d . This moduli space is top ologized as a subspa ce of the co nt inuous tw o fold lo op s pa ce, Rat d ( CP m ) ⊂ Ω 2 d CP m . This space 11 can b e describ ed as a configura tion space of ( d + 1)-tuples of complex po lynomials, z → ( p 0 ( z ) , p 1 ( z ) , · · · , p d ( z )) where the p i ’s are all monic p olyno mia ls of degree d that don’t shar e a common ro ot. By identif ying a monic p o ly nomial of degree d with its d ro ots, Segal considered this space o f rational functions as a certain configuration s pace of points in C (the configura tion of the ro ots of all the p olynomia ls), which then allow ed him to describ e gluing maps , Rat d ( CP m ) → R at d +1 ( CP m ) . The following theorem answers b oth Stabilit y Questio ns 1 and 2 in this setting. Theorem 13. (Se gal [54]) Both the gluing maps Rat d ( CP m ) → Rat d +1 ( CP m ) and the inclus ion maps Rat d ( CP m ) ֒ → Ω 2 d CP m ar e homotopy e quivalenc es t hr ough dimension d (2 m − 1) . F urthermor e b oth of t hese maps induc e monomorphisms in homol o gy in al l dimensions. Notice that all of the pa th co mpo nents of Ω 2 CP m are homotopy equiv alent to each other. This can b e s een by applying lo op m ultiplications by a map of degree one ι , and by a map o f degree − 1 j , × ι : Ω 2 d CP m → Ω 2 d +1 CP m × j : Ω 2 d +1 CP m → Ω 2 d CP m . Since ι and j are homotopy inv erse to each other , each of these ma ps is a ho motopy equiv a le nce. Then the ab ov e theo rem implies tha t if R at ∞ CP m is the (homo to p y) colimit lim d →∞ Rat d ( CP m ), then there is a homotopy eq uiv a lence Z × Rat ∞ ( CP m ) ≃ Ω 2 CP m . (2) W e remark that the homo topy type and esp ecially the homology of Ω 2 CP m is fairly well un- dersto o d. Studying the c a nonical circle bundle, S 1 → S 2 m +1 → CP m , yields, by an element ary homotopy ar gument, that Ω 2 S 2 m +1 → Ω 2 1 CP m is a homotopy equiv a lence. Said a nother wa y , Z × Ω 2 S 2 m +1 ≃ Ω 2 CP m . The to p o lo gy o f Rat d ( CP m ) was further studied by Co hen-Cohen-Mann-Milg ram in [12]. The stable homotopy type of these ra tional function space s was completely determined, and in particular their homo logies were calculated explicitly . The ca se of m = 1 is particularly interesting, co nsidering the fact that b oth Rat d ( CP 1 ) and the classifying s pace of the braid gro ups B β q give an a pproximation of the homology type of Ω 2 S 2 (compare T he o rem 13 and Theore m 12). In [1 2] the following was prov ed. 12 Theorem 14. Rat d ( CP 1 ) and B β 2 d have the same stable homotop y t yp e. In p articular they have isomorphi c homolo gies. Analogues of the sta bility Theorem 13 for rationa l functions with v alues in Gra ssmannians, o r more g eneral ho mogeneous spaces were proved by K ir wan, Guest, and Gr av esen in [3 9], [27], a nd [2 6]. In [15], Cohen, Jo nes, and Segal g av e a Morse theor etic pro of of Grav esen’s theorem, and studied the general question of when a clos ed, simply connected, in tegral symplectic manifold has this type of stability prop erty for its mo duli space o f (based) r ational maps (i.e. holomorphic maps fr o m CP 1 ). The explicit homology t ype of these more general r ational function spaces were co mputed by Boy er-Hurtubise-Mann-Milg ram in [8]. Segal also pr ov ed a stability r esult for spaces of holomo r phic maps from a higher g enu s Riemann surface to CP m . Let Σ g be a closed Riemann surface of gen us g , and let H ol d (Σ g , CP m ) be the spa c e of based holomorphic maps of genus g . Like in the ca se of rational functions, this is topo logized as a subspace of the space of contin uous based maps, M ap d (Σ g , CP n ). Theorem 15. (Se gal [54]) If g > 0 the inclusion H ol d (Σ g , CP m ) ֒ → M ap d (Σ g , CP m ) is a homolo gy e quivale nc e up t o dimension ( d − 2 g )(2 m − 1) . Again, it is easy to s ee that the homotopy type of M ap d (Σ g , CP m ) is independent of d , and so this theorem desc r ib e s the stable homolo gy t yp e of H ol d (Σ g , CP m ). Segal’s theorem can be e xtended to in volv e families o f complex structures on the surface Σ g . Namely , let M g,d ( CP m ) be the mo duli space of ho lomorphic curves o f genus g and deg ree d in CP m . More sp ecifically , M g ( CP m ) is defined as follows. Fix a smo oth, c lo sed, oriented s urface F g of genus g . Then M g ( CP m ) is the quo tien t space M g ( CP m ) = { ( J, φ ) , where J is an (almost) complex structur e on F g , and φ : ( F g , J ) → CP n is holomorphic o f degree d } / Diff + ( F g ) . Here Diff + ( F g ) is the space of or ient ation pres e r ving diffeomorphisms which acts diagona lly on the space of (almost) complex structures o n F g , a nd on the space of maps F g → CP m . One can also define a to po logical a nalogue, M top g,d ( CP m ) which is defined similar ly , except that φ : F g → CP n need only b e a co ntin uous map. Recently , D. Ayala prov ed the follo wing extension o f Segal’s theorem: Theorem 16. (Ayala [3]) The obvious inclusion M g,d ( CP m ) ֒ → M top g,d ( CP m ) induc es an isomorphism in homolo gy with c o efficients in a field of char acteristic zer o in dimensions less than ( d − 2 g )(2 m − 1) . 13 The hypothes is that the coefficient field hav e characteristic z e r o has to do with the fact that the action of the diffeomor phism gro up on the space of complex structures has stabilizer gr oups which are of the homoto py t yp e of finite gro ups. This hypothesis ca n be remov ed if o ne defined these mo duli spaces using the homotopy orbit spaces of the diffeomo rphism groups, rather than the actual orbit s paces. Equiv alen tly , one co uld define these mo duli spaces a s the quotien t sta ck of this action. This theor em can be viewed as addressing Stability Q uestion 1 in this setting. Co mb ined with a theorem of Cohen and Madse n [16], whic h gives an e x plicit calculatio n of H ∗ ( M top g,d ( X )), for X any simply connected spa c e , through dimension ( g − 5) / 2, this theo rem can also b e v iewed as addr essing Stabilit y Question 2. The Cohen-Mads e n result is closely related to, and uses in its pro of, the work of Harer and Iv ano v on the homologica l stabilit y of mapping cla ss groups [2 8], [37], and of Madse n and W eiss on their pro of of Mumford’s conjecture on the stable cohomo lo gy of the mo duli space of Riemann surfac e s [42]. These sta bilit y theorems will be discussed in more detail in Section 5. 3.2 Gauge theory 3.2.1 Flat conne ctions on Ri e mann surfaces In a seminal pap er [1 ], A tiyah and Bott studied the topo lo gy of the mo duli s paces of Y ang-Mills connections on Riemann surfac e s , and related them to mo duli spa ces of holomorphic bundles. W e will descr ibe one o f their main results, and interpret it as a stabilit y theo rem for these mo duli spaces. Let Σ b e a close d Riemann surface of genus g , a nd let E → Σ b e a principa l G -bundle, where G is a co mpact Lie group. T o make the statements of the following theorems easier, w e will assume that G is semisimple. Let g b e the Lie algebra , and L e t ad ( E ) = E × G g → Σ the c o rresp onding “adjoint bundle” , where G acts on g by conjugation. Let A ( E ) be the space of connections o n E , and let A F ( E ) b e the subspa ce c onsisting o f flat connections. In the semisimple setting, these flat connections minimize the Y ang-Mill functional, Y M : A ( E ) − → R Y M ( A ) = k F A k 2 where F A is the cur v a ture 2-form, and kk is the L 2 -norm on Ω ∗ (Σ; ad ( E )). Let G ( E ) be the gauge gro up of the bundle E . This is the group o f principal G -bundle a u- tomorphisms of E → Σ that live ov er the identit y map of Σ. The inclusion of flat connections , A F ( E ) ֒ → A ( E ) is a G -equiv ariant embedding, and the following is one of the ma in results of [1]. Theorem 17. (Atiya h-Bott). The inclusion A F ( E ) ֒ → A ( E ) induc es an isomorph ism on G -e qu ivariant homolo gy in dimensions less than 2 ( g − 1) r , wher e g is the genus of Σ , and r is the smal lest numb er of the form 1 2 dim ( G/Q ) , wher e Q ⊂ G is any pr op er, c omp act sub gr oup of maximal r ank. 14 Remark. The theorem is s tated in [1] in a slightly different for m. They obser ve that a c on- nection on E determines a holo morphic structure on the co mplexification E c . This allows for the ident ification of the space A ( E ) of connections on E with the spac e of ho lomorphic structur es on E c . They then show that there is a Morse-type G -equiv ariant s tratification of this space of holomorphic bundles, a nd they compute the rela tive codimensio ns o f the s trata. The s pace of flat connections is homotopy equiv alent to the s tratum of “s e mi-stable” ho lomorphic bundles, and the kno wledge of the codimensio n of this stra tum in the next low est stra tum (in the par tial order ing) leads to a simple calcula tion of the ( G -equiv ariant) c o nnectivity o f the inclusion A F ( E ) ֒ → A ( E ). See [13] for details of this calculation. Notice that this theor e m can be v iewed as a sta bilit y theorem since the range of equiv arian t homology isomorphism increases linearly with the genus. This can b e seen slightly mor e explicitly as fo llows. Recall tha t if X is a space with an action of a group K , its equiv ar iant ho mology , H ∗ K ( X ) is defined to b e the (ordina ry) homolog y o f its homotopy or bit space, X//K defined to b e E K × K X , where E K is a co ntractible space with a free K -action. W e ther efore may c onsider the follo wing homotopy or bit spaces of the gauge group action, M F ( E ) = A F ( E ) // G B ( E ) = A ( E ) // G . Theorem 17 can then be restated as follows. Corollary 18. The inclusion M F ( E ) ֒ → B ( E ) induc es an isomorphism in homolo gy in dimensions less than 2( g − 1 ) r . Notice fro m the ab ov e discussion, that this can be viewed as a sta tement a bo ut the homology o f the mo duli space of semistable holomor phic bundles over Σ o f the top ologica l t yp e of the complexified bundle, E c . Now rec a ll from [1] that B ( E ) is homotopy equiv alen t to the mapping space M ap E (Σ , B G ), where B G = E G/G is the classifying s pace of principal G - bundles , and M ap E represents the comp onent of the contin uous mapping space consisting of maps that clas sify bundles isomorphic to E . This mapping space ha s easily describ ed homo topy type (see [1 ]), so this interpretation of the Atiy ah-Bott theorem can b e vie w ed as an answer to Stability Question 2 in this setting. W e remark that the Atiy ah-Bott theo rem has been extended to allow the complex structure o n Σ to v ary ov er moduli space. This w as accomplished in [13]. The mo duli space under study in that work was defined to b e M G g,E = ( A F ( E ) × J (Σ)) // Aut( E ) , where Aut( E ) is the group of G -equiv ariant maps E → E which lie o ver some orientation preserving diffeomorphism Σ → Σ. By forg etting the bundle data ther e is a fibration sequence M F ( E ) → M G g,E p − → M g 15 where M g is the mo duli space J (Σ) // Diff + (Σ). The following w as prov ed in [13]. Theorem 19. [13] Ther e is a map α : M G g,E → M ap E (Σ , B G ) // Diff + (Σ) that induc es an isomorphi sm in homolo gy in dimensions less than 2( g − 1 ) r . F urther more the homo logy of M ap (Σ , B G ) // Diff + (Σ) ha s b een co mputed in dimensions les s than or e q ual to ( g − 4 ) / 2 explicitly b y Cohen and Madsen in [16]. In pa rticular its ra tional cohomolog y (in this range ) is freely gener ated by H ∗ ( B G ), and by the Miller-Mor ita-Mumford canonical classes κ i . Again, this r esult makes heavy use of Madsen and W eiss’s pro of o f Mumford’s conjecture, which will b e discuss e d further in section 5. 3.2.2 Self dual connections on four-manifol ds and the A tiy ah-Jones Co njecture One of the most imp orta nt gauge theor etic stability theor ems was pr ov ed by B oyer, Hurtubise, Mann, and Milgram [8]. This stability theorem had to do with the moduli spaces of self dual connec tio ns on S U (2 )- bundles ov er S 4 , and was a verification o f a well known conjecture of Atiy ah and Jones [2]. The setup for this theorem is the following. Isomorphism class es of pr incipal S U (2)-bundles over S 4 are cla ssified by their second C her n cla ss, c 2 ∈ H 4 ( S 4 ) ∼ = Z . Let p k : E k → S 4 be a pr incipal S U (2)-bundle with Cher n class k . Let A k be the space of c o nnections o n E k , and A k sd the subspace of self dual co nnections. Here we ar e g iving S 4 the usual round metric. A sd forms the space o f minima of the Y a ng-Mills functional, Y M : A k → R defined by Y M ( A ) = k F A k 2 , muc h like in the Riemann surfac e case . Let G k be the based gaug e gr oup of E k . This is the group of bundle automo rphisms g : E k → E k living ov er the identit y map of S 4 , with the pr op erty that o n the fib er over the basep oint ∞ ∈ R 4 ∪ ∞ = S 4 , g : ( p k ) − 1 ( ∞ ) → ( p k ) − 1 ( ∞ ) is the identit y . The gauge gro up G k acts freely on A k , so its or bit space B k = A k / G k is the classifying s pace of the gauge g roup. A stra ightf orward ho motopy theoretic arg umen t origina lly due to Gottlieb [25] s ays that there is a homotopy equiv alence, B k ≃ Ω 4 k B S U (2) where the subscr ipt denotes the comp onent of the spa ce of based maps γ : S 4 → B S U (2) with γ ∗ ( c 2 ) = k ∈ H 4 ( S 4 ). The fact that Ω B G ≃ G is true for any gro up G , implies that Ω 4 B S U (2) ≃ Ω 3 S U (2) = Ω 3 S 3 , and hence B k ≃ Ω 3 k S 3 . Now let M k ( S 4 ) = A k sd / G k be the mo duli space of self dual connections on E k . The inclusion A k sd ֒ → A k defines a map M k ( S 4 ) ֒ → B k ≃ Ω 3 k S 3 16 which w as studied by Atiy ah and J ones in [2]. Using solutions of the self-dual equations due to the physicist ’t Ho oft, Atiy ah and Jo nes were able to prove the following stability theorem in [2]: Theorem 20. [2] The map H ∗ ( M k ) → H ∗ (Ω 3 k S 3 ) is surjective for ∗ < k − 2 . A tiyah and Jones then made the following conjectures: 1. The inclusion M k ⊂ Ω 3 k S 3 is a homo logy isomor phism in dimensions t ≤ q ( k ) for some increasing function q ( k ) with lim k →∞ ( q ( k )) = ∞ . 2. The range o f the surjection (iso morphism) q = q ( k ) can b e explicitly determined as a function of k . 3. The homolo g y sta temen ts can be replac e d by homotopy statement s in both co njectur e s 1 a nd 2. The la st and stro ngest sta tement be c ame commonly known a s the Atiy ah-Jo nes conjecture. While it is ea sy to construct ma ps j k : Ω 3 k S 3 → Ω 3 k +1 S 3 , which ar e ho motopy equiv a lences, there was, a t the time, no o bvious analogous map g k : M k → M k +1 . Later, T aub es defined s uch gluing maps analytically [56]. In particula r he show ed that the following dia gram homotopy comm utes M k g k − − − − → M k +1   y   y Ω 3 k S 3 j k − − − − → Ω 3 k +1 S 3 . This diagr a m p ermits (homoto p y) direct limits and hence a stable version o f the At iyah-Jones conjecture. This was verified by T aub es in [55] by ana lytically studying the indices o f the nonminimal critical p oints of the Y ang-Mills functional. Theorem 21. [55] L et M ∞ b e the homotopy dir e ct limit of the M k ’s under the inclusions g k and let θ : M ∞ − → Ω 3 0 S 3 b e t he dir e ct limit of the inclusions M k ⊂ Ω 3 k S 3 . Then θ is a homotopy e quiva lenc e. Notice that this ca n b e viewed as an a nswer to Stabilit y Questio n 2 in this s e tting. The answer to Stabilit y Q ue s tion 1 w as supplied by Boyer, Hurtubise, Ma nn, and Milgra m with their pro o f of the Atiy ah-Jones conjecture in [8]. W e remark that one can ask the analogous types of stability questions when one studies con- nections on principal bundles for differe nt Lie gro ups , and o n differ e n t four dimensio nal manifolds. T a ubes prov ed the analog ue of Theore m 21 in this full generality . The full extent to which the analogue the Atiy ah-Jones conjecture ho lds is still an open ques tio n. 17 4 General linear groups, Pseud oisotopies, and K -theory 4.1 The st able top ology of general linear gr oups and algebraic K -theory Let R b e a discrete ring and GL n ( R ) the ra nk n -ge ner al linear group. Understanding the cohomolo g y of the group GL n ( R ) is imp ortant in algebra , top ology , algebraic geo metry , and num ber theory . One may vie w GL n ( R ) a s the subgroup of GL n +1 ( R ) co nsisting of matrices that have zer os in all entries of the ( n + 1) st row and ( n + 1) st column except the ( n + 1 ) × ( n + 1) entry , which is a 1 . This inclusion defines a map on cla ssifying spa ces, ι n : B GL n ( R ) → B GL n +1 ( R ). Let B GL ( R ) b e the (homotopy) direct limit of these maps. Recall that Quillen defined the algebraic K -groups, K i ( R ), to be the i th homotopy gr oup K i ( R ) = π i ( B GL ( R ) + ) , where Quillen’s plus construction is a very explicit cons truction that changes the homo topy t ype, but does not change the homolog y . In this co nt ext, the Stability Ques tions 1 and 2 were a ns wered by Char ney [10] in the case when R is a Dedekind domain, w hen s he proved the following. Theorem 22. [10] F or R a De dekind domain, t he induc e d maps ι n : H i ( B GL n ( R )) → H i ( B GL n +1 ( R )) ar e isomorphisms if 4 i + 5 ≤ n . If R is t he ring of inte gers in a numb er field, ι n : π i ( B GL n ( R ) + ) → π i ( B GL n +1 ( R ) + ) is an isomorp hism for 4 i + 1 ≤ n . Generalizations o f these homo logical stability theorems were found by Dwyer [18] and v an der Kallen [5 8]. The theorem was genera lized to wider classes of ring s, to c ertain clas ses of nontrivial co efficients mo dules, and the stability ranges were impr ov ed. 4.2 Pseudoisotopies, and W aldhausen’s algebraic K -theory of spaces Let M n be a smo oth, compact ma nifold, per haps with b oundary . The group of pseudoiso to pies P ( M ) is defined to b e the diffeomorphism group, P ( M ) = Diff ( M × I ; ∂ M × I ∪ M × { 0 } ) . This group natura lly acts on Diff ( M ) in the following wa y . Consider the ho momorphism P ( M ) → Diff ( M ) which maps H ∈ P ( M ) to H 1 ∈ Diff ( M ) defined to b e the restriction of H to M × { 1 } . The a ction of H on Diff ( M ) is g iven by H f = f ◦ H 1 . T wo diffeomor phisms f 1 and f 2 are s a id to b e pseudois otopic if they lie in the s ame orbit of this g roup action. Notice that f 1 and f 2 are isotopic if they lie in the same path component o f Diff ( M ). In a seminal pap er [11], Cerf addressed the question: “If f 1 and f 2 are pseudoisotopic, are they isotopic?”. In [11] Cerf pr ov ed the following: 18 Theorem 23. [11] L et M b e a simply c onne ct e d, C ∞ , close d, n - dimensional manifold with n ≥ 6 . Then P ( M ) is c onne ct e d. Ther efor e in this setting, pseudoisotopi c diffe omorphisms ar e isotopi c. The top olog y of the space of pseudoisoto pies has b een of great interest ever since that time. In particular, Hatcher and W ag oner [3 4] show ed that π 0 ( P ( M )) is not necessar ily tr ivial if M is not simply connected, ev en when n ≥ 6. W e will not sta te pre c isely the r esult of their ca lculations of π 0 ( P ( M )) here, but they are rela ted to the algebraic K -theor y of the g roup ring of the fundamental group, K ∗ ( Z [ π 1 ( M )]). There is a natura l “susp ension” map, σ : P ( M ) → P ( M × I ) defined by ess ent ially letting σ ( H ) b e H × id . W e say “essentially” b ecause a smoo thing pr o cess m ust b e done to deform H × id so that it satisfies the requisite b o undary conditions. Let P ( M ) = li m k →∞ P ( M × I k ) where the limit is a homotopy colimit under the maps σ . This spa c e of “stable pse udo isotopies” is of g reat interest, b ecause W a ldhausen prov ed that it is an infinite loop spa ce that can b e studied K -theor etically . In particular he defined the notion of the “ Algebraic K -theory of a space ” , A ( X ). (Here X can be any spa c e - not nece s sarily a manifold.) This is the algebr aic K -theory of the “ring up to ho motopy”, Q ((Ω X ) + ), where as ab ov e, Ω X is the loop space of X , and the co nstruction Q ( Y ) is as defined in section 1. The set of path c o nnected comp onents can be identified with the group ring, π 0 ( Q ((Ω X ) + )) ∼ = Z [ π 1 ( X )] and Q ((Ω X ) + ) can itself b e vie w ed as a t yp e of group ring in the a ppropriate catego ry of infinite lo op spaces. In a ny case, the following was one o f W aldha usen’s ma jor theorems ab out these spaces. Theorem 24. (Se e [61]) The sp ac e A ( X ) splits as a pr o duct of infinite lo op sp ac es, A ( X ) ≃ W h ( X ) × Q ( X + ) wher e W h ( X ) is r eferr e d to as the “Whitehe ad s p ac e” of X . In p articular if X is a manifold, W h ( X ) has as its two-fold lo op sp ac e, the sp ac e of stable pseudoisotopies, Ω 2 W h ( X ) ∼ = P ( X ) . Of course it then beca me very imp orta nt to understand how the spac e of stable pseudo isotopies P ( M ), which, by W aldhausen’s theor em can be studied K -theo retically , approximates the or iginal unstable gro up of pse udo isotopies P ( M ). Igus a’s stability theore m [36] answered this very impo rtant question. It can b e v iewed as a n answer to Stability Question 1 in this context, and together with W aldha usen’s theorem, we also hav e an answer to Sta bility Question 2 . Theorem 25. [36] (Igusa). The susp ension map σ : P ( M n ) → P ( M n × I ) induc es an isomorphism in homotopy gr oups in dimensions k so long as n > max (2 k + 7 , 3 k + 4 ) . 19 5 The mo duli space of Riemann surfaces, mapping class groups, and the Mumford c onjecture Probably the most bas ic , imp ortant mo duli spaces o ccurr ing in geometr y and top olog y are the mo duli spaces M g,n of genus g Riema nn surface s with n b oundar y comp onents. The ir top ology has bee n of cent ral interest since the 19 60’s, and has had importa n t applications to algebraic geometry , low dimensiona l top olog y , dyna mical systems, conforma l field theory and string theory in physics, and most re cent ly , algebra ic topolo gy . Recently , Madsen and W eiss [42] identified the “stable top ology ” of these mo duli spa ces, while proving a generalization of a fa mous conjecture of Mumford. In this sectio n we descr ibe some of the ing r edients of their stabilization theorems, as well as a related new theorem of Gala tius, a bo ut automorphisms of fre e g roups [21]. 5.1 Mapping class groups, mo duli spaces, and T hom spaces The mo duli spa ces M g,n can be defined as follows. Let Σ g,n be a fixed smo oth, compact, o riented surface of g enus g > 1, and n ≥ 0 bo undary comp onents. Let H g,n be the s pace of hyper bo lic metrics on Σ g,n with geo desic b oundary , such that each b oundary cir cle has length one. The moduli space is then defined to b e M g,n = H g,n / Diff + (Σ g,n , ∂ Σ g,n ) , where a s e a rlier, Diff + (Σ g,n , ∂ Σ g,n ) consists o f orientation preserving diffeo mo rphisms that ar e the ident ity on the b ounda ry . Re c a ll that T eichm¨ uller space, T g,n , can b e obta ined by taking the quotient of H g,n by the subg roup Diff + 1 (Σ g,n , ∂ Σ g,n ) of diffeomorphisms iso topic to the identit y . The quotient gro up Diff + (Σ g,n , ∂ Σ g,n ) / Diff + 1 (Σ g,n , ∂ Σ g,n ) is the discr ete gr oup of is otopy c la sses of diffeomorphisms, known as the mapping class group Γ g,n . This can be viewed as the group of path comp onents, Γ g,n = π 0 (Diff + (Σ g,n , ∂ Σ g,n )). In pa r ticular we then hav e M g,n = T g,n / Γ g,n . When the surface has b oundar y (i.e. n > 0), the actio n of the mapping class g roup on T g,n is free. Moreov er, since T g,n is homeomorphic to Euclidean spac e , T g,n ∼ = R 6 g − 6+2 n , the mo duli s pace is a classifying spa c e for the mapping cla ss group, M g,n ≃ B Γ g,n . (3) When the surface is closed, the action of Γ g,n on T g,n has finite stabilizer groups. This implies that for k any fie ld of characteristic zero, there is still a homology isomor phism, H ∗ ( M g, 0 ; k ) ∼ = H ∗ ( B Γ g, 0 ; k ) . (4) F urther more, since the subgro up Diff + 1 (Σ g,n , ∂ Σ g,n ) is contractible, o ne also has that the full diffeo- morphism gro up Diff + (Σ g,n , ∂ Σ g,n ) has contractible comp onents. This implies that the pro jection 20 on its comp onents Diff + (Σ g,n , ∂ Σ g,n ) → Γ g,n is a ho motopy equiv alence, and hence there is an equiv alence of classifying spaces, B Diff + (Σ g,n , ∂ Σ g,n ) ≃ B Γ g,n . (5) Putting this equiv alence together with equiv alence (3) we s ee that for n > 0, the mo duli s pace M g,n is homotopy equiv alent to B Diff + (Σ g,n , ∂ Σ g,n ), and therefor e c lassifies smo oth Σ g,n -bundles. W e no w assume n ≥ 1, and we consider group homomor phisms, σ 1 , 0 : Γ g,n → Γ g +1 ,n , and σ 0 , − 1 : Γ g,n → Γ g,n − 1 (6) defined as follows. Pick a fix e d b oundar y c ir cle c ⊂ ∂ Σ g,n Consider an embedding e g,n : Σ g,n ֒ → Σ g +1 ,n that sends all o f the ( n − 1) b ounda ry circles of ∂ Σ g,n other than c diffeomor phica lly to bo undary circles of ∂ Σ g +1 ,n , and so that Σ g +1 ,n = Σ g,n ∪ c T where T is a surface o f genus one w ith t w o b oundar y c ircles, c and c ′ . In other words, Σ g +1 ,n is obtained fro m Σ g,n by g luing in the surface of genus one, T . Given an isotopy class o f diffeomor- phism of Σ g,n , γ ∈ Γ g,n , the elemen t σ 1 , 0 ( γ ) ∈ Γ g +1 ,n is the isotopy class de fined b y e x tending a diffeomorphism in the class of γ to all o f Σ g +1 ,n by letting it b e the identit y o n T ⊂ Σ g +1 ,n . This defines the ho momorphism σ 1 , 0 : Γ g,n → Γ g +1 ,n . The map σ 0 , − 1 is defined similarly . Namely one c ho oses a n embedding κ g,n : Σ g,n ֒ → Σ g,n − 1 that se nd all of the ( n − 1) b oundary circles of ∂ Σ g,n other than c diffeomorphically to the n − 1 bo undary circles of ∂ Σ g,n − 1 , and so that Σ g,n − 1 = Σ g,n ∪ c D where D is diffeomorphic to the disk D 2 . In other w ords, Σ g,n − 1 is obtained from Σ g,n by “capping off” the b oundar y circle c ∈ ∂ Σ g,n by attaching a disk. By extending a repr e sentativ e diffeomorphism of a cla ss γ ∈ Γ g,n by the iden tit y on D ⊂ Σ g,n − 1 , one obtains a homomorphism σ 0 , − 1 : Γ g,n → Γ g,n − 1 . Notice that the ho momorphisms σ 1 , 0 and σ 0 , − 1 depe nd on the isotopy cla sses of the choice of embeddings e g,n and κ g,n resp ectively , but the following famous theo r em of Harer [2 8] shows that any such c hoice induces an isomorphism in homology through a ra nge. This re sult can be viewed as an answer to Stablit y Q uestion 1 in this context: Theorem 26. [28][37] F o r g > 1 and n ≥ 1 , t he homomorphisms, σ 1 , 0 and σ 0 , − 1 induc e isomor- phisms in the homol o gy of the classifying sp ac es, σ 1 , 0 : H q ( B Γ g,n , Z ) ∼ = − → H q ( B Γ g +1 ,n , Z ) σ 0 , − 1 : H q ( B Γ g,n , Z ) ∼ = − → H q ( B Γ g,n − 1 , Z ) for 2 q < g − 2 . 21 Remarks. 1. Hare r’s o riginal theo r em did not have as la rge a stability range as describ ed here. This ra nge is due to Iv a nov [37]. The stabilit y r ange has b een impro ved even further by Boldsen [6]. 2. Notice that this re s ult holds for n = 1. It therefo re implies that the homology of the mapping class gr o ups for closed surfaces , H q ( B Γ g, 0 ), is indep endent of g so lo ng as 2 q < g − 2. 3. This theor em was generalize d to include certain families of twisted co efficients by Iv a nov [3 8], Cohen-Madsen [16], and with improv ed stabilit y rang es b y Boldsen [6]. Combining this theo r em with sta temen ts (3) and (4) above, one has the following c orollar y . Corollary 27. F or g > 1 and n ≥ 1 , the homolo gy of the mo duli sp ac e of R iemann surfac es H q ( M g,n ; Z ) is indep endent of g and n so long as 2 q < g − 2 . This r esult hold s for t he mo duli sp ac e of close d surfac es M g, 0 as wel l, if one takes homolo gy with c o efficients in a field k of char acteristic zer o. These res ults can b e viewed as a nswering Stability Ques tio n 1 in the case o f the mo duli space of curves. One of the ma jo r re c en t adv ances o f the sub ject was the answering of Stabilit y Questio n 2 in this setting by Madsen and W eiss [42] when they pro ved a generalizatio n of a long sta nding conjecture of Mumfor d [49] ab out the stable c o homology of mo duli space, o r equiv alently , of the stable cohomolo gy of mapping class gr oups. A wa y of stating Mumford’s co njectur e is a s follows. Let B Γ ∞ ,n be the mapping telesco p e (homotopy colimit) of the maps o n classifying spaces B Γ g,n σ 1 , 0 − − → B Γ g +1 ,n σ 1 , 0 − − → B Γ g +2 ,n σ 1 , 0 − − → · · · By the Har er stability theorem, H ∗ ( B Γ ∞ ,n , Z ) is indep endent of the n umber of b oundar y comp onents n . Moreover it is iso morphic to the homolo gy of the “ infinite genus mo duli space”, H ∗ ( M ∞ ,n ; Z ) for n ≥ 1, and if o ne takes co efficients in a field k of characteristic zero, this homology is iso morphic to the ho mology of the clos e d ma pping class group H ∗ ( M g, 0 ; k ) if the ge nus g is lar ge with resp ect to the homolog ical degree. Mumford’s co njectur e was ab out the stable cohomolog y H ∗ ( B Γ ∞ , 1 ; k ) where k is a field of characteristic zero: Conjecture 28. (Mumfor d) [49] The stable c ohomolo gy of t he mapping class gr oups is a p olynomial algebr a, H ∗ ( B Γ ∞ , 1 ; k ) ∼ = k [ κ 1 , κ 2 , · · · , κ i , · · · ] wher e κ i ∈ H 2 i ( B Γ ∞ , 1 ; k ) is t he Mil ler-Morita-Mumfor d c anonic al class. The Miller-Mor ita-Mumford c la sses [49], [4 8], [46] can b e defined in the following wa y . As remarked ab ove (5) there is an equiv alence o f c la ssifying spaces, B Γ g,n ≃ B Diff + (Σ g,n , ∂ Σ g,n ), and so these space s class ify surfa ce bundles whose structure group is this diffeomor phism gr oup. In 22 particular the coho mology of these classifying spa ces is the algebra of characteristic classes o f such bundles. So let Σ g,n → E → B be a smo oth bundle with Diff + (Σ g,n , ∂ Σ g,n ) as its str ucture group. Consider the vertical tangent bundle, T ver t E → E . The fib er at y ∈ E , consists of thos e ta ngent v ectors in T y E that are tangent to the fib er surface at y of E → B . This bundle is a t wo dimensiona l, or iented vector bundle (recall the structure gro up Diff + (Σ g,n , ∂ Σ g,n ) cons ists of o rientation preser ving diffeomor phisms). Let e ∈ H 2 ( E ) b e its E uler class. T he n the κ -cla sses are defined by int egra ting pow ers of e a long fiber s, κ i = Z fib er e i +1 ∈ H 2 i ( B ) . Alternatively , this is the pushforward in coho mology , κ i = p ! ( e i +1 ). Beca us e of the naturality of the pushforward (integration) construction, these classes define characteris tic classes , and therefore lie in H ∗ ( B Diff + (Σ g,n , ∂ Σ g,n )). Since their constructio n did not de p end on the g enus of the surface, they actually define s ta ble cohomo logy classes in H ∗ ( B Γ ∞ , 1 ). W e re ma rk that these classes can be constructed directly in H ∗ ( M g,n ), by integrating a lo ng fib ers as ab ove, in the canonical Σ g,n -bundle, Σ g,n → M 1 g,n → M g,n where M 1 g,n is the mo duli spa ce of curves with one marked point. W e r emark that Miller prov ed in [46] that the induced map k [ κ 1 , κ 2 , · · · , κ i , · · · ] → H ∗ ( B Γ ∞ , 1 ; k ) is injective. How ev er to prove Mumford’s conjecture (that it is an isomorphism), Madsen and W eiss employ ed metho ds of homotop y theo ry a s well as differential top ology . I n [57], Tillmann considered the Quillen plus constr uc tio n, applied to the mapping class groups, B (Γ g,n ) + , and their stabilization, B Γ + ∞ , 1 . As men tioned e a rlier, this construction do es not alter the homolog y , so understanding the homotopy t yp e of B Γ + ∞ , 1 would yield an unders tanding of the stable cohomolo g y of mapping cla ss groups. In [57], Tillmann prov ed that B Γ + ∞ , 1 is an infinite lo o p spa ce. This result is similar in spirit to Q uillen’s result that B GL ( R ) + is an infinite lo op space (used to define higher algebra ic K - theory). In homotopy theory , infinite lo o p spaces define generalized cohomolog y theo r ies, but it wasn’t clear what g eneralized co ho mology theory B Γ + ∞ , 1 defined. Using a homo topy theoretic mo del of integrating a lo ng fib ers tha t stems from Pont rjagin and Tho m’s famous work on c o b o rdism theory , Madsen conjectured what the coho mology theory w as. This conjectur e w as studied by Ma dsen and Tillmann in [41], and was even tually proved by Madsen and W eiss in [42]. Once this co homology theory was identified, Mumfor d’s conjecture was an immediate c o nsequence, as was a des cription, in principle, of the s ta ble coho mology , H ∗ ( B Γ ∞ , 1 ; Z ) with integer co efficients. The cohomo logy H ∗ ( B Γ ∞ , 1 ; Z / p ) was later computed explicitly by Ga latius in [20]. The answer is quite complicated, but it is en tirely defined in terms o f rather standard o b jects in homotopy theory (“Dyer-Lashof op erations” ). 23 A ba sic ingredient in the Madse n-W eiss pro o f is the use of the Pont rjagin- T ho m construction to give a ho motopy theoretic mo del for “integrating along fibers ”. In this par ticula r setting, Madsen and W eiss use a different model of the classifying space B Diff + (Σ g ). (Here I am considering closed surfaces Σ g , but there is a n a nalogous constructio n for surfaces with b oundar y , that is eq ually treated in [42].) As describ ed in the discussion on Whitney’s embedding theorem (3), the space E mb (Σ g , R ∞ ), is a co n tractible space , with a free actio n of Diff + (Σ g ). W e then hav e that the quotient space, E mb (Σ g , R ∞ ) / Diff + (Σ g ) ≃ B Diff + (Σ g ). This is the moduli space o f s ubsurfaces of R ∞ of genus g , which w e denote by S g ( R ∞ ). Consider the spa c e of subsurfaces of R N , S g ( R N ) = E mb (Σ g , R N ) / Diff + (Σ g ). There is an obvious Σ g -bundle ov er S g ( R N ), Σ g → S 1 g ( R N ) p − → S g ( R N ) , where S 1 g ( R N ) is the spa ce of subsurfa ces of R N with a marked point. Now c onsider the ma p p × ι : S 1 g ( R N ) → S g ( R N ) × R N where ι ( S, x ) = x ∈ R N . This is an e mbedding . By identifying R N with B R (0), the ba ll of radius R a round the origin, we can co nsider a n induced embedding S 1 g ( R N ) ֒ → S g ( R N ) × B R (0) ⊂ S g ( R N ) × R N . Let η b e a tubular neighborho o d. More sp ecifically this em bedding has a normal bundle, ν N , which over eac h sur face ( S, x ) ∈ S 1 g ( R N ), is the orthogonal complement of the tang en t space T x S in R N . One can then extend this embedding to an embedding of an ǫ -neig hborho o d of the zero section of ν N , for sufficient ly small ǫ > 0 . The image of this embedding is the tubular neighborho o d η . One then has a “ Pon trjagin-Thom collapse” map τ : S g ( R N ) × R N / ( S g ( R N ) × ( R N − B R (0))) − → S g ( R N ) × R N / (( S g ( R N ) × R N ) − η ) . The left hand side can b e identified with the N -fold susp ension, Σ N ( S g ( R N ) + ), and the right hand side can b e identified with the Thom space of the norma l bundle ν N . Notice that this is an or iented, ( N − 2)-dimensional vector bundle. An easy bundle theor etic exer cise shows that the Whitney sum, ν N ⊕ T ver t S 1 g ( R N ) ∼ = S 1 g ( R N ) × R N (7) viewed as the N -dimensional tr ivial bundle. W e ca n therefore think of ν N as the “vertical no rmal bundle” of the pr o jection map p : S 1 g ( R N ) → S g ( R N ). F urther mo re this isomo rphism a nd the orientation o f T ver t S 1 g ( R N ) induces a n orientation o n ν N . The Pon trjagin-Thom map can then b e v iewed as a map τ : Σ N ( S g ( R N ) + ) → T hom ( ν N ) , (8) where we are using the notation T hom ( ζ ) to denote the Thom spa ce of a vector bundle ζ . 24 Since ν N is oriented there is a Thom isomo rphism, H q ( S 1 g ( R N )) ∼ = − → H q + N − 2 ( T hom ( ν N )), a s well as a susp ension iso morphism, H j ( S g ( R N )) ∼ = − → H j + N (Σ N ( S g ( R N ) + )). With r esp ect to these isomorphisms, the induced cohomology homomor phism defined b y the Pontrjagin-Thom map, τ ∗ : H ∗ ( T hom ( ν N )) → H ∗ (Σ N ( S g ( R N ) + )) induces a homo morphism, H q ( S 1 g ( R N )) → H q − 2 ( S g ( R N )) which is well known to be equal (up to sign) to the fib erwise in tegration map (or pushforw ard ma p) p ! : H q ( S 1 g ( R N )) → H q − 2 ( S g ( R N )). This homotopy theor etic view of fiber wise integration has, in some sens e a universal mo del. Namely , if Gr + 2 ( R N ) is the Grass mannian of oriented 2-dimensio nal s ubspaces o f R N , notice that there is a na tural map j : S 1 g ( R N ) → Gr + 2 ( R N ) ( S, x ) → T ver t,x S 1 g ( R N ) Notice tha t the vertical tangent space a t x is a subs pa ce of the tangen t space, which in tur n is a subspace o f R N since S ⊂ R N . Notice furthermore, that by definition, if γ 2 ,N → Gr + 2 ( R N ) is the canonical, or iented 2 - dimensional bundle, then j ∗ ( γ 2 ,N ) = T ver t S 1 g ( R N ) . (Recall γ 2 ,N consists of pairs, ( V , v ), where V ⊂ R N is an oriented, 2-dimensio nal subspace, a nd v ∈ V .) Let γ ⊥ 2 ,N be the orthogo nal complement bundle. This is the ( N − 2)-dimensio nal bundle ov er Gr + 2 ( R N ) that cons is ts of pa irs ( V , w ), where V ⊂ R N is a n o riented, 2-dimens io nal subspa c e, a nd w ∈ V ⊥ . The bundle equation (7) induces a n isomo r phism, j ∗ ( γ ⊥ 2 ,N ) ∼ = ν N . F urther more j induces a ma p of Thom space s , j : T hom ( ν N ) → T hom ( γ ⊥ 2 ,N ) . The a djoint of the Pontrjagin-Thom map τ : Σ N ( S g ( R N ) + ) → T hom ( ν N ), is a map τ : S g ( R N ) → Ω N ( T hom ( ν N )), and if we comp ose with the map j , we obta in a map α g,N : S g ( R N ) → Ω N ( T hom ( γ 2 ,N ) ⊥ ) . Now as observed in [42] there ar e na tural inclus io ns Ω N ( T hom ( γ 2 ,N ) ⊥ ) ֒ → Ω N +1 ( T hom ( γ 2 ,N +1 ) ⊥ ) that are compa tible with the inclusions S g ( R N ) ֒ → S g ( R N +1 ). W e wr ite Ω ∞ ( T hom ( − γ 2 )) as the (homotopy) dir e c t limit of these maps. In the langua ge of homotopy theory , this is the z e r o space of the Thom sp ectr um of the virtual bundle − γ 2 , where γ 2 → Gr + 2 ( R ∞ ) is the cano nical or i- ent ed 2- dimensional bundle. Notice that this can b e identified with the canonica l co mplex line bundle L → CP ∞ , and so Ω ∞ ( T hom ( − γ 2 )) can b e identified with Ω ∞ ( T hom ( − L )). (Besides the 25 notation given here, ther e ar e s everal “standard” notations for this infinite lo o p space, including Ω ∞ (( CP ∞ ) − L ), Ω ∞ ( CP ∞ − 1 ), and mo re recently , Ω ∞ M T S O (2).) In any ca se, by pa ssing to the limit one has a map α g : B Diff + (Σ g ) ≃ S g ( R ∞ ) → Ω ∞ T hom ( − γ 2 ) . (9) The following is the Madsen-W eiss theorem, which supplies a dramatic answer to Stability Ques- tion 2 in this setting. Theorem 29. [42] The maps α g define d ab ove extend to a map α : Z × B Γ + ∞ , 1 → Ω ∞ T hom ( − γ 2 ) which is a homotop y e quivalenc e (of infinite lo op sp ac es). In p articular, the stable c oh omolo gy of the mapping class gr oups, H ∗ ( Z × B Γ ∞ , 1 ; G ) is isomorphic to H ∗ (Ω ∞ T hom ( − γ 2 ); G ) for any c o effici ent gr oup G . As men tioned above, the homotopy type of Ω ∞ T hom ( − γ 2 ) is rather complicated, but it is a natural ho motopy theor etic constr uc tio n, whose basic ingredient is the canonica l line bundle L → CP ∞ . In particular the rational cohomology calc ulation is rather easy , and it is easily s e e n to imply Mumford’s co njectur e (28). An imp or tant implication of this homotopy equiv alence and Galatius’s calculation [21], is that the stable cohomo logy of the mapping class gr o ups (and the mo duli spaces of curves) has a rich to r sion comp onent that classical geometric techniques did not detect. Aside from the P ontrjagin-Thom co ns truction, the main idea in the Madsen-W eiss pro o f w as to give a geometric interpretation of the statemen t in the ab ov e theorem. This was done by comparing concorda nc e (cob ordis m) clas ses of sur fa ce bundles, M n +2 → X n , which ar e class ified by B Diff + ( F g ) for some g , with concorda nce class es of smoo th pro pe r maps q : M n +2 → X n that come equipp ed with bundle epimo r phisms, δ q : T M × R i → q ∗ ( T X ) × R i that live ov er q : M → X . Notice that no assumption is made that the bundle map δ q is related to the differ e ntial dq . Pontrjagin-Thom theory says that as i gets large, this latter set of concor dance clas ses of maps is classified by Ω ∞ T hom ( − γ 2 ). The compa rison of these tw o sets of concordance clas ses of maps was studied using an “ h -principle” prov ed b y V assilie v [59]. A more detaile d outline of the metho ds used by Madsen and W eiss is contained in the introduction to their paper [42]. A significant simplification of the pro of of the Madsen-W eiss theorem w as achiev ed recently by Galatius-Madsen- Tillmann-W e is s [22]. This pap er is ab out the top olog y of “cob ordism categorie s”. An n -dimensional cob or dis m categor y has ob jects consisting o f closed ( n − 1)-ma nifolds, and its morphisms are n -dimensional cob ordisms b etw een them. These ma nifolds may carry prescr ibed structure on their tangent bundles, suc h as orient ations, almost complex structures, or framings. Much care is given in [22] to give pr e cise definitions to these top ologic a l categories. Such cob ordism categorie s, aside from the relev ance to the stable top ology of moduli s paces, also are relev an t in 26 studying top ologica l quantum field theories, a nd hence their top olo gies (i.e. the top olo gy of their classifying spa c es) is of gr eat in terest. In [57] Tillmann proved that Z × B Γ + ∞ , 1 has the sa me homotopy t yp e as the classifying space of the 2-dimensional oriented cob ordism category Ω B C ob or 2 . Her pr o of inv olved a clever use of Harer’s stability Theorem 26 [28], a nd a modificatio n o f the “g r oup completion” tec hniques o f McDuff a nd Segal [45]. In this remark able pap er , the four authors o f [22] then iden tified the homotopy type of the classifying space of any such cob ordism category , in any dimension. T ogether with Tillmann’s theorem, this gave a simplified pro of of the Madsen-W eiss theo rem. Moreov er, this theorem, being prov ed in the generality it was, has had significant influence on the sub ject b eyond the study of the mo duli space of curves. Cob ordism theory has b een central in differential top olo gy s ince the original works of Pontrjagin and Thom. The r esults of this pap e r follow the spirit of Thom’s classification of cob or dism classes of manifolds, but they go further. The pap er gives a coherent way of studying the cob ordis ms that defined the equiv a lence relation in Thom’s theor y . This work has inspired considerable work b y many people in algebr aic and differential topolo gy ov er the last few years. Unfortunately , the des cription of muc h of this new work is b eyond the scop e of this pa per . 5.2 Automorphisms of free groups One last stabilit y pheno menon that w e will discuss concer ns a utomorphisms of the free g roup on n -genera tors, Aut ( F n ), and the outer a utomorphism g r oups, Out ( F n ), defined to b e the quotient Ou t ( F n ) = Aut ( F n ) /I nn ( F n ), where I nn ( F n ) < Aut ( F n ) is the subgroup of inner automor phisms. The stability theorems regarding these groups run pa r allel to, b oth in s ta temen t, and to a certain extent in pro o f, to the s ta bilit y theor ems reg a rding mapping class groups of sur faces due to Har er a nd Madsen-W eiss, des crib ed ab ov e. In particular , wherea s the mapping class g roup Γ g,n is the g r oup of isotopy classes o f diffeomorphisms of a s urface, the automor phism gr oup Aut ( F n ) is the group of (based) homotopy cla sses o f (based) homoto p y equiv alences o f a graph G n , who s e fundamental group is the free gr oup on n -genera tors. Similarly Ou t ( F n ) ca n b e viewed as the gro up of un based homotopy clas ses of unbased homotopy equiv alences of G n . In [17] Culler and V og tmann describ ed a simplicial co mplex whose simplices ar e indexed b y graphs having fundament al gr oup F n . This space ha s a natural action of Out ( F n ) and in many wa ys is analo gous to T eichm¨ uller space, with its a ction of the mapping class g roup. This space b ecame known as “Outer Space” , and ha s led to many imp ortant ca lculations. In par ticular, if o ne quotients by the ac tion of O ut ( F n ) one is studying the mo duli space of gr aphs, and it is shown to have the same rational homology as the classifying spac e , B O ut ( F n ). This should be viewed as the a nalogue of the re la tionship b etw een the moduli space of curves and the classifying space o f the mapping class group (3) (4). Now the natur a l inclusion F n < F n +1 defines a map ι n : Aut ( F n ) → Aut ( F n +1 ). W e us e the same nota tion fo r the induced map of cla ssifying spa ces, ι n : B Aut ( F n ) → B Aut ( F n +1 ). Similarly , the pro jection maps p n : Aut ( F n ) → O ut ( F n ) define maps o n classifying s paces, p n : B Aut ( F n ) → 27 B O ut ( F n ). The following stabilit y theorem of Hatc her and V og tma nn was pr ov ed in [31], [32], [33]. It ca n b e view ed as the analog ue of Harer ’s stability Theore m 2 6 ab ove, and can a lso b e view ed as an answer to Stabilit y Q uestion 1 rega rding the moduli space of gr aphs. Theorem 30. [31],[3 2 ],[33]. The induc e d maps in homolo gy, ι ∗ : H i ( B Aut ( F n ); Z ) → H i ( B Aut ( F n +1 ); Z ) and ( p n ) ∗ : H q ( B Aut ( F n ); Z ) → H q ( B O ut ( F n ); Z ) ar e isomorphisms for 2 i + 2 ≤ n and 2 q + 4 ≤ n r esp e ctively. This theorem naturally leads to the problem of co mputing s ta ble ho mology of groups Aut ( F n ). In [2 9] Hatcher c o njectured that the rational stable homolog y is zero. More pr ecisely , let Aut ( F ∞ ) be the direct limit of the groups, Aut ( F ∞ ) = lim n →∞ Aut ( F n ) . Conjecture 31 . [29](Hatcher) The r a tional homolo gy gr oups, H i ( B Aut ( F ∞ ); Q ) = 0 for i > 0 . This c o njecture was rece n tly prov ed, in dramatic fashion, by S. Ga latius in [21]. Galatius actually prov ed a theorem that computes this stable homolog y with any co efficie n ts. Theorem 32. [21](Galatius) L et Σ n b e t he symmetric gr oup on n -letters. View Σ n as the sub gr oup of Au t ( F n ) given by p ermutations of the gener ators of F n . Th en the map on classifyi ng sp ac es B Σ n → B Aut ( F n ) induc es an isomorphism in homolo gy, H i ( B Σ n ; G ) ∼ = − → H i ( B Aut ( F n ); G ) for 2 i + 2 ≤ n , and G any c o efficient gr o up. In p articular t he induc e d map B Σ ∞ → B Aut ( F ∞ ) is a homolo gy e quivalenc e. Whe n one applie s the Q uil len plus c onstruction, ther e ar e homotopy e quiva lenc es, Z × B Σ + ∞ ≃ − → Z × B Aut ( F ∞ ) + ≃ − → Ω ∞ S ∞ wher e, like ab ove, Ω ∞ S ∞ = lim n →∞ Ω n S n . 28 Notice tha t, since the symmetric groups ar e finite, they ha ve triv ial r ational ho mology . Thus Hatcher’s conjecture is a corolla r y of Galatius’s theorem. No w the ho mology of the sy mmetric g roups is completely kno wn with any field co efficients, and hence the stable homolog y of the automor phism groups of free gr oups is similarly now k nown. Notice this result is co mpatible with the B arratt- Priddy-Quillen Theore m 11 regar ding the stable homolo gy of symmetric groups. Galatius’s a rgument is s imilar to the Madsen-W eiss argument in s pir it, but inv olved man y new ideas and co nstructions. A key idea in Galatius’s ar gument is to build a mo del for B O u t ( F n ) as a space of gra phs embedded in E uclidean space. This builds on the Culle r -V o gtmann mo del of “Outer Space”. He then defined a “ scanning pro cedure”, muc h like wha t was used b y Sega l in his study o f rational functions [5 4], to define a map α : B O ut ( F n ) → Ω ∞ Φ where Ω ∞ Φ is the natural home for the image of a P ontrjagin-Thom type map. (Notice that this is not the standard Pontrjagin-Thom constuction, since the graphs inv olved are ob viously not smo o th manifolds.) This space is itself defined fro m a shea f of (noncompac t) graphs. The induced maps B Aut ( F n ) → Ω ∞ Φ then extends to a map α : B Au t ( F ∞ ) + → Ω ∞ Φ. This map is analo gous to the Madsen-Tillmann map α : B Γ + ∞ , 1 → Ω ∞ T hom ( − γ 2 ) describ ed in Theorem 29 ab ov e. Galatius then prov ed that this map is a ho motopy equiv a lence. Finally he prov ed that Ω ∞ Φ has the homotopy t yp e of Ω ∞ S ∞ . W e re ma rk that Galatius’s method of pro of is quite general, and in particular leads to a further simplification of the pro of of the Ma dsen-W eis s theo rem, a s well as the theorems of [22] on cob ordism categorie s. It has also lead to considerable generaliza tions of these theorems (see [4][24][23]). 6 Final Commen ts The stability theo rems considered he r e come in different types, and ha ve a v ariety of different characteristic features. How ever they hav e all had a significa nt impact on their field of research, and in some cas e s that impact has b een quite dramatic. It therefor e seems that it would be quite v a luable to understand the common features of the clas sifying spa ces a nd moduli spaces that admit stability theo r ems, and to tr y to understand common feature s of their pro o fs. F or example, ma n y o f the stability theo r ems co nsidered he r e hav e to do with cla s sifying s pa ces o f sequences of groups. They included braid groups, symmetric g roups, genera l linear groups, mapping class gro ups, a nd automor phis ms of free g roups. The common wa y in whic h Stability Question 1 was pr ov ed in these cases ha s b een considered by Hatcher and W ahl. In a ll of these ca ses, simplicial complexes with the appropriate g roup actions w ere found or constructed, with certain criteria on the stabilizer subgroups. Then, typically , a spectra l sequence a rgument was used to inductiv ely prov e a stabilty theo r em. Understa nding the g eneral prop erties of these group actions (i.e. finding axioms) that would imply these sta bilit y theor ems has lead to the discov ery of new such theorems (see for example, [35], [60]). 29 As for other gener al features, no tice tha t some o f the ab ove stabilit y theore ms that addr essed Stabilit y Question 2 inv olved some v ariation o f the Pontrjagin-Thom construction. This was true of the pro ofs o f Theorems 8, 13, 29, 3 2 describ ed ab ov e. It would certainly be of great v alue to un- derstand under wha t conditions the Pontrjagin-Thom co nstruction yields a (homology) equiv alence, and therefore a s ta bilit y theor em. Some of the stability theorems describ ed a bove a re, in a sense, mor e analytic in nature. They concern the moduli space of solutions to a differen tial equation, such as the Cauch y-Riemann equa- tion in the case of ho lomorphic curves, o r self-duality eq uations in the cas e of Y ang- Mills moduli spaces. In these cases , b oth Stabilit y Questions 1 and 2 inv olve unders tanding the relative top ology of the mo duli spa ce of solutions inside the e ntire configuration space (e.g. holomo rphic curves inside all smoo th cur ves, or self-dual co nnections inside all connections). In [15] Cohen, Jones, and Segal discussed sufficien t Morse-theo retic conditio ns on w he n the space of rational maps to a s ymplectic manifold approximate the top ology of all contin uous maps of S 2 to the manifold. Their condition inv olv ed a kind of homog e neit y pro pe r ty . How ever it is far from unders to o d, in gene r al, for what t yp e of symplectic manifolds, and for what c hoices of compatible almo st complex structure is there a stability theorem for spaces of pseudo-holomorphic curves. Stabilit y theor ems hav e b een impor ta n t in both algebraic a nd differential top olo gy , a s well as bo th alge br aic and differential geometry . 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