On real algebraic realization of round fold maps of codimension $-1$

The canonical projections of the unit spheres are generalized to special generic maps and round fold maps, for example. They are generalizations from the viewpoint of singularity theory of differentiable maps and these maps restrict the topologies and the differentiable structures of the manifolds. We are concerned with round fold maps, defined as smooth maps locally represented as the product map of a Morse function and the identity map on a smooth manifold, and maps with singular value sets being concentric spheres. A bit different from differential topology, we are concerned with real algebraic geometric aspects of these maps. We discuss real algebraic realization of round fold maps of codimension $-1$ as our new work. Real algebraic realization of these maps is of fundamental and important studies in real algebraic geometry and a new study recently developing mainly due to the author.

💡 Research Summary

The paper “On Real Algebraic Realization of Round Fold Maps of Codimension –1” by Naoki Kitazawa investigates the problem of representing a class of smooth maps, called round fold maps, by real algebraic maps. Round fold maps are a special subclass of fold maps: locally they look like the product of a Morse function with the identity on a manifold, and globally their singular value set consists of a finite collection of concentric spheres in the target Euclidean space. The most elementary example is the restriction of the standard projection π_{m+1,m−1}: ℝ^{m+1}→ℝ^{m−1} to the unit sphere S^{m}. Such maps have been studied extensively in differential topology because they impose strong restrictions on the topology and smooth structure of the source manifolds.

The author’s main goal is to show that, under natural topological conditions, these round fold maps can be realized as restrictions of the same linear projection to the zero set of a suitable real polynomial. In other words, the smooth map f : M→ℝ^{m−1} is A‑equivalent (i.e., C^∞‑equivalent up to diffeomorphisms of source and target) to a map of the form π_{m+1,m−1}|_{Z(F)} where Z(F)⊂ℝ^{m+1} is the real algebraic hypersurface defined by a polynomial F. This bridges the gap between differential topology and real algebraic geometry for this class of maps.

The paper proceeds in several stages:

1. Background on Fold and Round Fold Maps.
   The author reviews the definition of fold maps, the index i(p) of a singular point, and the special generic case i(p)=0. Round fold maps are defined precisely as fold maps whose singular set is embedded and whose image consists of concentric spheres S_i={x∈ℝ^n | Σ x_j^2 = i^2} for i=1,…,ℓ. The notion of A‑equivalence and R‑equivalence (when the target diffeomorphism is the identity) is introduced, together with the concept of a page function ˜f_{P,x₀} obtained by restricting a Morse function to a ray emanating from a point on the unit sphere.

2. Reeb Graphs, SM‑Digraphs, and Classification Results.
   The paper adopts the language of Reeb graphs (quotient spaces of a map by connected components of level sets) and introduces SM‑digraphs (simple Morse digraphs) to encode the combinatorial data of Morse functions on compact manifolds. The author recalls several classification theorems (Theorems 2–4) originally proved with Saeki: for m≥5, a round fold map on a closed orientable m‑manifold is uniquely determined up to A‑equivalence by the R‑equivalence class of its page function; conversely, any suitable Morse function on a surface yields a round fold map. Moreover, explicit topological characterizations are given for manifolds admitting round fold maps whose image is a disk D^{m−1} or a product S^{m−2}×D^1, in terms of connected sums of S^1×S^{m−1} or products S^{m−1}×S where S is a surface of prescribed genus.

3. Trivial Spinning Construction (TSC).
   A central technical tool is the “trivial spinning construction”. Starting from a Morse function (the page function) on a compact (m−n)-manifold, one forms the product with the identity on S^{n−1} and, if necessary, glues in a trivial bundle D^{n}×C where C is a disjoint union of circles. The resulting map is a round fold map, called a TSC map. The author shows that all round fold maps appearing in the classification theorems are, up to A‑equivalence, TSC maps.

4. Main Realization Theorem (Theorem 0 and its refinements).
   The core contribution is Theorem 0, which asserts that for an integer m>3, any round fold map f : M→ℝ^{m−1} satisfying one of the following conditions can be realized algebraically:

   • (1) M = S^{m−2}×S where S is a closed connected orientable surface, and either m>4 or (m=4 and the singular set S(f) minus the set of definite fold points is non‑empty.
   • (2) M is orientable and f(M) is diffeomorphic to the (m−1)-disk D^{m−1}.

   In both cases, there exists a real polynomial F such that π_{m+1,m−1}|_{Z(F)} is A‑equivalent to f. The proof proceeds by first constructing a TSC map with the prescribed page function, then approximating the Morse function by a real polynomial (using Whitney C^∞‑topology and generic position arguments) so that the zero set of the polynomial inherits the required smooth structure and the projection’s singular set becomes exactly the prescribed concentric spheres.

5. Implications and Future Directions.
   By providing explicit algebraic models for round fold maps, the paper demonstrates that the differential‑topological constraints imposed by such maps are compatible with real algebraic structures. This opens the possibility of studying further invariants (e.g., Betti numbers, characteristic classes) within the algebraic category, and suggests extensions to higher codimensions or to other classes of singular maps (e.g., stable maps with more complicated singular sets). The interplay between SM‑digraph combinatorics and real algebraic geometry highlighted here may inspire new techniques for constructing algebraic maps with prescribed singularities.

Overall, Kitazawa’s work offers a thorough synthesis of singularity theory, Morse‑theoretic graph techniques, and real algebraic geometry, delivering a concrete answer to the long‑standing question of whether round fold maps of codimension –1 admit real algebraic realizations. The results enrich both fields: differential topologists gain explicit algebraic representatives of their maps, while real algebraic geometers obtain new families of hypersurfaces whose projection maps have highly controlled singular behavior.