A hyperbolic approach to exp_3(S^1)

A h yp erb oli approa h to exp 3 ( S 1 ) S. C. F. Rose No v em b er 20, 2018 Abstrat In this pap er w e in v estigate a new geometri metho d of studying exp k ( S 1 ) , the set of all non-empt y subsets of the irle of ardinalit y at most k . By onsidering the irle as the b oundary of the h yp erb oli plane w e are able to use its group of isometries to determine expliitely the struture of its rst few onguration spaes. W e then study ho w these onguration spaes t together in their union, exp 3 ( S 1 ) , to repro v e an old theorem of Bott as w ell as to oer a new pro of (follo wing that of E. Sh hepin) of the fat that the em b edding exp 1 ( S 1 ) ֒ → exp 3 ( S 1 ) is the trefoil knot. 1 In tro dution and Preliminaries Sine the mid 1900s, the study of nite subsets of top ologial spaes has ropp ed up in one guise or anotherinitially b y Borsuk [Bor49 ℄, later orreted b y Bott [Bot52 ℄ and more reen tly b y Handel [Han00 ℄, T uey [T uf02℄ and Mosto v o y [Mos04 ℄. A few dieren t denitions ha v e b een giv en in the literature (whi h are all easily seen to b e equiv alen t), but the one that w e will use for this pap er is the follo wing. Denition 1.1 Let X b e a top ologial spae, and let exp k ( X ) denote the set of all non-empt y subsets of X of ardinalit y at most k . That is exp k ( X ) := { S ⊆ X | 0 < | S | ≤ k } There is a natural surjetiv e map π : X k ։ exp k ( X ) whi h tak es ( x 1 , . . . , x k ) to { x 1 , . . . , x k } ; w e th us endo w exp k ( X ) with the quotien t top ology indued b y this map. It is of ourse lear that under this top ology , exp 1 ( X ) ∼ = X . There are a n um b er of results that an b e pro v en ab out this spae and its relation to X . First of all, the map X 7→ exp k ( X ) is a homotop y funtor from U , the ategory of top ologial spaes and on tin uous maps, to itself. More preisely , w e ha v e the follo wing theorem. 1 Theorem 1.2 L et X , Y b e top olo gi al sp a es, let f , g : X → Y . If F : X × I → Y is a homotopy fr om f to g , then the map F : exp k ( X ) × I → exp k ( Y )  { x 1 , . . . , x k } , t  7→ { F ( x 1 , t ) , . . . , F ( x k , t ) } is a homotopy fr om exp k ( f ) to exp k ( g ) . F rom whi h it immediately follo ws that Corollary 1.3 The homotopy typ e of exp k ( X ) dep ends only on that of X . There are also natural inlusions exp k ( X ) ֒ → exp m ( X ) for k ≤ m , taking the subset { x 1 , . . . , x k } to { x 1 , . . . , x k } ; th us w e ha v e X ∼ = exp 1 ( X ) ֒ → exp 2 ( X ) ֒ → · · · ֒ → exp k ( X ) ֒ → · · · with ea h exp k ( X ) for X Hausdorem b edded as a losed subset of exp m ( X ) . The pro ofs of these results (as w ell as man y other top ologial and homotop y- theoreti results) are in [Han00 ℄, whi h pro vides a go o d in tro dution to man y of the basi prop erties of exp k ( X ) (whi h he denotes as S ub ( X , k ) ). It should also b e noted that m u h of this is in mark ed on trast to the notion of a onguration spae, whi h is not funtorial (unless w e suitably restrit our ategory); nor are there nie em b eddings as ab o v e. The most famous result in the study of these spaes, ho w ev er, omes from Bott's orretion to Borsuk's initial pap er. That is: Theorem 1.4 The sp a e exp 3 ( S 1 ) is home omorphi to S 3 , the thr e e-spher e. This has b een pro v en so far in a n um b er of dieren t w a ysBott uses an elab orate ut-and-paste st yle argumen t (in eet sho wing that it the union of tori together with a alulation of its fundamen tal group to sho w that it is a simply onneted lens spae), T uey nds a deomp osition of it as a ∆ -omplex (in the sense of [Hat02 ℄) and sho ws that it is a simply onneted Seifert-b ered spae, while Mosto v o y ties it in with a result of Quillen ab out latties in the plane. There are of ourse man y other results ab out similar su h spaes, primarily in [T uf02 ℄ and [KS℄, most notably the follo wing result. Theorem 1.5 The homotopy typ e of exp k ( S 1 ) is that of an o dd dimensional spher e; sp e i al ly, exp 2 k ( S 1 ) ≃ exp 2 k − 1 ( S 1 ) ≃ S 2 k − 1 This is pro v en in [KS ℄ using the notion of trunated pro dut spaesthese an b e onsidered to b e, in a ertain sense, the free Z / 2 set on p oin ts in our spae. More sp eially , these are giv en b y T P n ( X ) := S P n ( X ) / ∼ 2 where S P n ( X ) is the n -th symmetri pro dut of X dened to b e the olletion of unordered n -tuples in X that is, S P n ( X ) = { x 1 + · · · + x n | x i ∈ X } and w e sa y that x 1 + · · · + x k − 2 + 2 x ∼ x 1 + · · · x k − 2 (where w e use the on v en tion of writing an unordered olletion of p oin ts as a sum), top ologized aordingly . The reason that this is useful is that T P n ( S 1 ) ∼ = RP n (see [Mos98 ℄) and so if w e then note the homeomorphism exp k ( X ) / exp k − 1 ( X ) ∼ = T P k ( X ) / T P k − 2 ( X ) this then b eomes exp k ( S 1 ) / exp k − 1 ( S 1 ) ∼ = RP k / RP k − 2 W e are then able to use the long exat sequene of homology groups to sho w that exp k ( S 1 ) is a homology spheresine it is simply onneted it is then homotopi to a sphere (b y a standard argumen t). Moreo v er, the fat that the exp funtor is a homotop y funtor has also b een used in [T uf03a ℄ and [T uf03b ℄ to b egin the determination of the homotop y t yp e of exp k (Σ) for Σ a losed surfae. The idea is that if y ou onsider a puntured surfae, then this is homotopi to a w edge of opies of S 1 and from the understanding of exp k ( S 1 ) T uey is then able to pro v e results ab out exp k (Γ ℓ ) , where Γ ℓ is a graph onsisting of w edges of ℓ irles. F rom this, he uses a Ma y er-Vietoris argumen t, o v ering a surfae with k + 1 opies of X i = Σ \ { p i } (for distint p oin ts p i ), to determine information ab out the homology of the spae exp k (Σ) (this w orks sine T k +1 i =1 X i = Σ \ { p 1 , . . . , p k +1 } , and so these o v er exp k (Σ) ). In this pap er w e in tend to pro vide y et another pro of Theorem 1.4 , b eginning with ertain prop erties of P S L 2 ( R ) . This metho d will generalize and pro vide metho ds of alulation not only of the homotop y t yp e of higher dimensional analogues, but p ossibly ev en their homeomorphism t yp e. The main dierene in this approa h as ompared to previous ones lies in its inheren t geometri nature. This hea vy emphasis on the underlying geometry of the irle (more generally of higher dimensional spheres) and its onnetion to h yp erb oli spae aords us a m u h more sp ei understanding of the top ology of the resulting spae. In on trast, previous metho ds (for higher dimensional analogues) w ere restrited to kno wledge of the homology and homotop y groups of these spaes whi h p ermitted a kno wledge of the homotop y t yp e of these spaes and no more. Sp eially , w e use the top ology of the group of isometries of the h yp erb oli plane H  P S L 2 ( R ) whi h w e note is homeomorphi to H × S 1 via the map T 7→  T ( i ) , dT dz ( i )  (1) Using this and the fat that S 1 = ∂ H , w e then mo d out ertain subgroups whi h p ermits us to determine the homeomorphism t yp e of exp 3 ( S 1 ) as laimed. 3 In this ase, as in previous pro ofs (e.g. [T uf02℄), w e tak e adv an tage of the ma hinery of Seifert bred spaes. It is quite easy to see that exp k ( S 1 ) has an S 1 ation on it giv en b y ζ · { x 1 , . . . , x k } = { ζ x 1 , . . . , ζ x k } and so, giv en that exp 3 ( S 1 ) is a losed 3-manifold (lemma 2.6 ), w e an use w ell kno wn lassiation theorems to determine the homeomorphism t yp e of exp 3 ( S 1 ) . Notation W e will use the notation C k ( X ) := { S ⊆ X | | S | = k } to denote the k -th unordered onguration spae of X . It is of ourse lear that exp k ( X ) = k [ j =1 C j ( X ) (2) The topi of this pap er will largely b e onerned with ho w this union b eha v es top ologially . 2 The analysis of exp 3 ( S 1 ) 2.1 The top ology and geometry of C i ( S 1 ) The main struture of the pro of of Theorem 1.4 is as follo ws. F or an y three p oin ts of S 1 = ∂ H = R ∪ ∞ there is an elemen t of P S L 2 ( R ) whi h tak es the p oin ts to { 0 , 1 , ∞} , preserving yli ordering. This elemen t is not unique; p ost-omp osing it with an y elemen t p erm uting { 0 , 1 , ∞} yields another p ossible  hoie. If w e let Γ b e the subgroup of P S L 2 ( R ) generated b y all elemen ts p erm uting { 0 , 1 , ∞} , then this subgroup ats on P S L 2 ( R ) b y left m ultipliation; if w e quotien t out b y this ation then w e obtain the follo wing result: Lemma 2.1 The sub gr oup of P S L 2 ( R ) whih yli al ly p ermutes (0 , 1 , ∞ ) is simply the yli gr oup Γ = Z / 3 , gener ate d by γ = z − 1 z ; thus C 3 ( S 1 ) ∼ = P S L 2 ( R ) / Γ By similar reasoning w e ha v e that for an y t w o p oin ts of S 1 there is an elemen t of P S L 2 ( R ) taking those p oin ts to { 0 , ∞} ; if w e then let Ξ b e the subgroup of P S L 2 ( R ) whi h xes the set { 0 , ∞} w e similarly ha v e the follo wing: Lemma 2.2 The sub gr oup of P S L 2 ( R ) whih setwise xes { 0 , ∞} is gener ate d by the elements τ = − 1 z and σ λ = λz (for λ ∈ R > 0 ) subje t to the r elations τ σ λ τ = σ − 1 λ = σ λ − 1 (3a) σ λ σ µ = σ λµ (3b) τ 2 = 1 (3) 4 whih le aves us with C 2 ( S 1 ) ∼ = P S L 2 ( R ) / Ξ And of ourse, C 1 ( S 1 ) is simply the irle S 1 . This tells us whi h subgroups w e need to b e mo dding out b y; what w e need to do next is study ho w these subgroups at on P S L 2 ( R ) so that w e ma y understand the top ology of the resulting quotien ts. Prop osition 2.3 C 2 ( S 1 ) , desrib e d as the quotient P S L 2 ( R ) / Ξ ab ove, is home- omorphi to the op en Möbius b and M . Pr o of W e ha v e that Ξ is generated b y τ and the σ λ , sub jet to the relations ab o v e. Note in partiular that b y (3a) it sues to onsider the ations of σ λ and τ separately . W e onsider the ation of σ λ rst. No w w e ha v e b oth ( σ λ ◦ T )( i ) = λ  T ( i )  d ( σ λ ◦ T ) dz ( i ) = λ dT dz ( i ) and so the ation of σ λ on P S L 2 ( R ) ∼ = H × S 1 is giv en b y σ λ · ( z , θ ) = ( λz , θ ) . If w e then quotien t out b y the ation of the subgroup of Ξ generated b y the σ λ 's w e nd P S L 2 ( R ) / h σ λ i ∼ = (0 , π ) × S 1 where the angle φ ∈ (0 , π ) is simply arg( z ) . Mo ving on to the ation of τ , w e note that ( τ ◦ T )( i ) = − 1 T ( i ) d ( τ ◦ T ) dz ( i ) = 1 T 2 ( i ) dT dz ( i ) and so sine up to saling (as an elemen t of (0 , π ) ) − 1 T ( i ) = π − φ , and similarly 1 T 2 ( i ) = − 2 φ , w e nd that P S L 2 ( R ) / Ξ ∼ = (0 , π ) × S 1 / ( φ, θ ) ∼ ( π − φ, θ − 2 φ ) whi h an b e pitorially represen ted as in gure 1 , whene follo ws the onlusion. Q.E.D. W e next mo v e on to the ase of C 3 ( S 1 ) . Prop osition 2.4 C 3 ( S 1 ) , desrib e d as a the quotient P S L 2 ( R ) / Γ ab ove, is home omorphi to the (op en) mo del Seifert br eing with twist 2 π / 3 . 5 θ φ 3 3 2 2 4 4 1 1 ∼ = ∼ = 2 π 0 π Figure 1: The homeomorphism b et w een C 2 ( S 1 ) and M Pr o of Sine γ = 1 − 1 z , w e ha v e immediately that ( γ ◦ T )( i ) = 1 − 1 T ( i ) (4) d ( γ ◦ T ) dz ( i ) = 1 T 2 ( i ) dT dz ( i ) (5) and so γ do es the same to the S 1 o ordinate as τ . No w, in the H o ordinate, it turns out that 1 − 1 z an b e written as R 1 ◦ R 2 where R 1 ( z ) = 1 − z R 2 ( z ) = z | z | 2 and so it is the omp osition of t w o reetions, rst ab out the irle | z | = 1 and then ab out the line ℜ z = 1 2 it is th us a rotation of the angle 4 π 3 ab out the p oin t e π i / 3 . With gure 2 as a fundamen tal domain, and with left and righ t edges iden tied aording to equations (4 ) and (5 ) , w e qui kly see that this is a mo del Seifert-breing as laimed. Q.E.D. 2.2 T op ology of the union exp k ( S 1 ) A t this p oin t w e ha v e a solid grasp as to exatly what the three piees C i ( S 1 ) , i = 1 , 2 , 3 are. Remaining still is to desrib e the top ology of their unionthat is, what happ ens when p oin ts b egin to oalese. W e b egin b y lo oking at exp 2 ( S 1 ) . First, in M ∼ = C 2 ( S 1 ) , w e nd that a pair { p, r } with p > r maps to the elemen t ξ ( z ) = z − p z − r 6 1 − 1 z z 0 + i 0 1 + i 0 e π i/ 3 D × S 1 Figure 2: Choie of F undamen tal Domain in P S L 2 ( R ) (with a sligh t mo diation if p, r = ∞ ), and so using the homeo- morphism from equation (1) w e see rst that ξ ( i ) = i − p i − r dξ dz ( i ) = p − r ( i − r ) 2 and so that as p → r , ξ ( i ) → 1 , while dξ dz ( i ) = p − r ( i − r ) 2 , sine w e an ignore saling, sta ys onstan t. F rom this the follo wing is immediate. Prop osition 2.5 The sp a e exp 2 ( S 1 ) = C 1 ( S 1 ) ∪ C 2 ( S 1 ) is home omorphi to the lose d Möbius b and M ∗ . Q.E.D. W e no w mo v e on to the situation with three p oin ts. First of all, our triple { p, q , r } (with p < q < r , without loss of generalit y) maps to the elemen t of P S L 2 ( R ) giv en b y ξ ( z ) =  q − r q − p  z − p z − r whi h, desrib ed in terms of equation (1) yields ξ ( i ) =  q − r q − p  i − p i − r (6) dξ dz ( i ) =  q − r q − p  p − r ( i − r ) 2 (7) and so as in the ase of t w o p oin ts, the p osition in the S 1 o ordinate only dep ends on a single one of these p oin tsin this ase, r . 7 T o aid in our desription, w e will assume that it is the p oin t q whi h tends to w ards either p or r . That b eing the ase, it should b e noted that as q v aries, the quan tit y q − r q − p v aries b et w een −∞ and 0 . As su h (see equation (6)), b y xing b oth p and r w e trae out a path in a partiular H slie whi h happ ens to b e a straigh t line from the p oin t 0 + i 0 of slop e p − r 1+ pr . Letting q → r (for simpliit y's sak e) w e approa h either 0 + i 0 or 1 + i 0 , dep ending on the sign of p − r 1+ pr (Consider our  hoie of fundamen tal domain). Then the Möbius band is glued on to C 3 ( S 1 ) en tirely at the p oin ts 0 + i 0 and 1 + i 0 . Ho w ev er, the p oin t whi h w e end up at on the Möbius band dep ends on the slop e of the path w e tak e to w ards these p oin tsw e are blo wing up these p oin tsand so w e nd that these p oin ts are in fat separated b y the set of lines passing through them, whi h are then glued to the orresp onding p oin t on the op en Möbius band. But what ab out C 1 ( S 1 ) = ∂ ( M ∗ ) , and what of the edge running from 0 + i 0 to 1 + i 0 ? The rst thing to note is that as the union exp 3 ( S 1 ) m ust b e ompat, an y sequene { p i , q i , r i } whi h tends to that edge m ust neessarily on v erge to an elemen t of C 1 ( S 1 ) in the union. The only question remaining is what that elemen t is. Let { p i , q i , r i } b e a sequene in exp 3 ( S 1 ) whi h on v erges (regarded as a p oin t in the quotien t H × S 1 / Γ ) to some p oin t P = ( λ + i 0 , θ ) where λ ∈ (0 , 1) , θ ∈ [0 , π ) . Cho ose a neigh brouho o d N of P su h that N is ompletely on tained in the fundamen tal domain of gure 2 . As su h, ea h { p i , q i , r i } is ev en tually on tained in this neigh b ourho o d, and th us in our partiular  hoie of fundamen tal domain. Ho w ev er, b y our  hoie of fundamen tal domain there is an expliit ordering no w on these p oin ts, and th us the H slie that an y { p i , q i , r i } =  ( p i , q i , r i )  nds itself in is determined b y a partiular one of these p oin ts, sa y r i . As this on v erges to w ards ( λ + i 0 , θ ) , w e m ust ha v e that r i → θ (as w ell as p i , q i ). That is, the p oin t in C 1 ( S 1 ) that w e on v erge to is simply the p oin t that all of p i , r i , q i are on v erging to. Th us in exp 3 ( S 1 ) , the edge (0 + i 0 , 1 + i 0 ) × { θ } simply ollapses to {∗} × { θ } . 2.3 exp 3 ( S 1 ) and the inlusion exp 1 ( S 1 ) ֒ → exp 3 ( S 1 ) W e will no w pro v e (after a qui k lemma) the main result of this pap er, a stronger v ersion of Theorem 1.4 (pro v en also in [ T uf02℄). Lemma 2.6 The sp a e exp 3 ( S 1 ) is a  omp at 3 -manifold without b oundary. Pr o of Compatness is immediate as exp 3 ( S 1 ) is a quotien t of ( S 1 ) 3 . As for it b eing a manifold, the only plae where this migh t fail is on exp 2 ( S 1 ) ⊂ exp 3 ( S 1 ) , so w e simply need to v erify that ea h p oin t therein has a eulidean neigh b ourho o d. F or p oin ts in C 2 ( S 1 ) this is rather easy . A p oin t in a neigh b ourho o d of { p, q } for p 6 = q will b e in one of the ongurations sho wn in gure 3, and so it is fairly easy to see that there is a neigh b ourho o d of { p, q } whi h is homeomorphi to 8 p q p q Figure 3: A neigh b ourho o d of { p, q } t w o opies of ( − ǫ, ǫ ) × C 2  ( − ǫ, ǫ )  glued along their ommon b oundarywhi h is then homeomorphi to ( − ǫ, ǫ ) 3 as required. F or a p oin t in C 1 ( S 1 ) , it is similarly easy to see that it has a neigh b ourho o d homeomorphi to exp 3  ( − ǫ, ǫ )  whi h onsidered as a quotien t of the spae X = { ( x, y , z ) | − ǫ < x ≤ y ≤ z < ǫ } (See gure 4 with faes A , B inden tied) is also readily seen to b e homeomorphi to a eulidean ball. A B Figure 4: exp 3  ( − ǫ, ǫ )  Q.E.D. Theorem 2.7 The sp a e exp 3 ( S 1 ) is home omorphi to S 3 , and the inlusion S 1 ∼ = exp 1 ( S 1 ) ֒ → exp 3 ( S 1 ) ∼ = S 3 is the tr efoil knot. 9 Pr o of The ma jorit y of this pro of will rely on alulating the fundamen tal groups of exp 3 ( S 1 ) and exp 3 ( S 1 ) \ exp 1 ( S 1 ) , relying on lassiation theorems to sho w the ab o v e result. The main to ol will b e the Seifert-V an Kamp en theorem. F rom the o v ering sho wn in gure 5, the Seifert-V an Kamp en theorem yields the follo wing pushout of groups. π 1 ( A ∩ B ) π 1 ( B ) π 1 ( A ) π 1  exp 3 ( S 1 )  / / i ∗   j ∗   / / where A deformation retrats on to a irle (and so π 1 ( A ) ∼ = h s i ), and B de- formation retrats on to M ∗ ≃ S 1 (hene π 1 ( B ) ∼ = h t i ); th us w e ha v e that π 1  exp 3 ( S 1 )  ∼ = h s, t | R i . It remains to determine what the relations R are. A B M ∗ e π i/ 3 Figure 5: The o v ering used for the Seifert-V an Kamp en theorem No w, A ∩ B is simply the b oundary of C 3 ( S 1 ) ; that is, up to homotop y it is simply a torus T 2 . Th us its fundamen tal group is h a i ⊕ h b i where a is the generator in the S 1 diretion, and b is the meridional generator. W e an expliitely desrib e this homotop y torus in terms of p oin ts on the underlying irle S 1 in the follo wing manner. The longitudinal diretion (ie the one orresp onding to a ab o v e) is giv en, as exp eted, simply b y rotation of p oin ts along S 1 ; as w e are a v oiding the exeptional bre, there is nothing un usual here and w e need a full rotation at an y giv en p oin t to return to the starting p oin t. No w, w e obtain the other generator b (demonstrated in gure 6) simply b y rotating ea h p oin t in a oun ter-lo  kwise manner to the next p oin t along. It is easy to see that this omm utes with a , and that together these t w o paths mak e a torus. 10 Figure 6: The generator b of π 1 ( A ∩ B ) So to determine the relations R , w e rst examine i ∗ ( a ) and j ∗ ( a ) , the simplest of the t w o to deal with. No w, as the generator of π 1 ( A ) is the path along the exeptional bre from θ = 0 to θ = π 3 , it is easy to see that j ∗ ( a ) = s 3 . The exat same reasoning sho ws that i ∗ ( a ) = t 2 , and so w e ha v e the relation that s 3 = t 2 . So for the meridional generator, b , w e ha v e the situation sho wn in gure 7 (f. gure 1) whi h an easily b e seen to b e homotopi to the generator t of π 1 ( B ) ; th us it only remains to see what happ ens to j ∗ ( b ) (sho wn in gure 8 ) to fully understand what π 1  exp 3 ( S 1 )  is. While it w ould b e tempting to suggest that it simply ollapses to a homotopially trivial path, this is not indeed the ase. ≃ i ∗ ( b ) Figure 7: The homotop y lass of i ∗ ( b ) This is easily seen geometrially; if w e p erturb that diagram of gure 6 so that our three p oin ts are equally spaed ab out the irle, then it is easy to see that the desription of the path b is exatly the path whi h generates π 1 ( A ) . Com bining all of this together, w e nd that π 1  exp 3 ( S 1 )  ∼ = h s, t | s 3 = t 2 , s = t i ∼ = 1 11 j ∗ ( b ) e π i/ 3 Figure 8: The homotop y lass of j ∗ ( b ) and so exp 3 ( S 1 ) is simply onneted. F rom [ST80 ℄ it follo ws immediately that as a simply onneted Seifert bred spae, this m ust b e homeomorphi to S 3 . F or brevit y w e will no w dene X := exp 3 ( S 1 ) \ exp 1 ( S 1 ) . No w, for the alulation of π 1 ( X ) the ma jorit y of the details ab o v e still hold throughthe only dierene is that what w as lab elled as B ab o v e (no w to b e denoted B ′ ) no longer deformation retrats on to M ∗ . Claim: π 1 ( B ′ ) ∼ = h t, u | [ t 2 , u ] = 1 i , where t is the generator of π 1 ( M ) and u is simply the image of the meridional generator in B ′ . F rom this laim it follo ws that π 1 ( X ) ∼ = h s, t, u | [ t 2 , u ] = 1 , s 3 = t 2 , s = u i ∼ = h s, t | [ t 2 , s ] = 1 , s 3 = t 2 i ∼ = h s, t | s 3 = t 2 i No w, to see that this implies that the inlusion exp 1 ( S 1 ) ֒ → exp 3 ( S 1 ) is the trefoil knot, w e pro eed as follo ws (This argumen t is due to E. Sh hepin). The rst thing is to note that the en ter of the Möbius bandits exeptional breis unk otted in exp 3 ( S 1 ) . This is due to the fat that π 1  C 3 ( S 1 )  ∼ = Z . Th us if w e onsider a tubular neigh b ourho o d of this subset w e obtain a torus in exp 3 ( S 1 ) and the in tersetion of the b oundary of this torus with the Möbius band is th us a torus knot whi h is isotopi to exp 1 ( S 1 ) . W e an no w use the fundamen tal group to sa y that it is a (2 , 3) torus knot, or a trefoil knot as laimed. Pro of of laim: Let us examine B ′ a little more losely . Figure 9 sho ws a slie of B ′ , separated in to op en sets U, V . No w, U ≃ T 2 , and V ≃ M . Lastly , U ∩ V ≃ S 1 , and so w e end up with the follo wing pushout to alulate π 1 ( B ′ ) : 12 U V Figure 9: Co v ering of B ′ h t i h a i ⊕ h b i h c i π 1 ( B ′ ) / / t 7→ a   t 7→ c 2   / / from whi h it follo ws that π 1 ( B ′ ) ∼ = h a, b, c | [ a, b ] = 1 , c 2 = a i ∼ = h b, c | [ c 2 , b ] = 1 i as laimed; Clearly , c is the generator of π 1 ( M ) , and b is the meridional gener- ator and so the full laim follo ws. Q.E.D. Referenes [Bor49℄ K. Borsuk. 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