[Paper Review] Notes on explicit special generic maps into Eulidean spaces whose dimensions are greater than 4

Naoki Kitazawa|arXiv (Cornell University)|Oct 20, 2020

Topological and Geometric Data Analysis25 references4 citations

TL;DR

This paper constructs explicit families of special generic maps from closed, simply-connected manifolds into Euclidean spaces of dimension ≥5, demonstrating that the integral homology and cohomology rings of the source manifolds are highly constrained. The key result shows that for certain dimensions and topological types, the cohomology subalgebra generated by torsion-free elements is isomorphic to that of $S^4 \times S^4$, with explicit control over Betti numbers and cup product structures.

ABSTRACT

Special generic maps are higher dimensional versions of Morse functions with exactly two singular points, characterizing spheres topologically except 4-dimensional cases and 4-dimensional standard spheres. The class of such maps also contains canonical projections of unit spheres. This class is interesting from the viewpoint of algebraic topology and differential topology of manifolds. These maps have been shown to restrict the topologies and the differentiable structures of the manifolds strongly by Calabi, Saeki and Sakuma before 2010s, and later Nishioka, Wrazidlo and the author. So-called exotic spheres admit no special generic map in considerable cases and homology groups and cohomology rings are shown to be strongly restricted. Moreover, special generic maps into Euclidean spaces whose dimensions are smaller than or equal to 4 have been studied well. The present paper mainly concerns cases where the dimensions of targets are greater than or equal to 5.

Motivation & Objective

• To construct explicit families of special generic maps from closed, simply-connected manifolds into Euclidean spaces of dimension at least 5.
• To analyze the topological and differentiable constraints imposed by such maps on the source manifolds, particularly on their integral homology and cohomology rings.
• To determine the structure of the subalgebra of cohomology classes of infinite order, showing it is isomorphic to that of $S^4 \times S^4$ in key cases.
• To extend known results on special generic maps beyond dimension 4, especially for exotic spheres and homotopy spheres.
• To provide explicit constructions and topological invariants (Betti numbers, cup products) for manifolds admitting such maps.

Proposed method

• Constructs families of closed, connected manifolds $\{M_r\}_{r \in \mathbb{Z}}$ and special generic maps $f_r: M_r \to \mathbb{R}^n$ for $n \geq 5$, satisfying specific dimension and topological constraints.
• Analyzes the image of the singular set $f_r(S(f_r))$, showing it is diffeomorphic to a manifold obtained by removing a tubular neighborhood of a submanifold in $S^{k_1} \times D^{k_2}$, with $k_1 < n/2$.
• Computes the integral homology groups $H_j(D_r; \mathbb{Z})$ of the image $D_r$, showing nontrivial groups only in specific degrees: $j = 0, n-1$, and $j = k_1, k-k_1, n-k-1, n-k+k_1-1, k, n-k_1-1$.
• Proves that the cohomology subalgebra generated by elements of infinite order is isomorphic to $H^*(S^4 \times S^4; \mathbb{Z})$, using Poincaré duality and the structure of the mapping cylinder.
• Applies techniques from [27] and uses the Euler characteristic of the source manifold $M$, computed as $\chi(M) = 4$, to constrain Betti numbers.
• Uses smooth isotopy arguments to show that certain 4-dimensional submanifolds (e.g., $C_{S(f)}$) can be isotoped away from each other, preserving the map structure.

Experimental results

Research questions

• RQ1 What are the topological constraints on closed, simply-connected manifolds that admit special generic maps into $\mathbb{R}^n$ for $n \geq 5$?
• RQ2 How do the integral homology and cohomology rings of such manifolds depend on the dimension and structure of the singular set?
• RQ3 Can the cohomology subalgebra generated by torsion-free classes be explicitly identified, and is it isomorphic to that of $S^4 \times S^4$?
• RQ4 What is the role of the Euler characteristic and Poincaré duality in constraining the Betti numbers of manifolds admitting such maps?
• RQ5 Under what conditions can the singular set and associated submanifolds be smoothly isotoped while preserving the special generic map structure?

Key findings

• For $n \geq 5$, there exist uncountably many families of closed, connected manifolds $\{M_r\}$ admitting special generic maps $f_r: M_r \to \mathbb{R}^n$ with prescribed singular set structure.
• The image $D_r = f_r(S(f_r))$ has integral homology $H_j(D_r; \mathbb{Z}) \cong \mathbb{Z}$ for $j = 0, n-1$, and trivial for $0 < j < k_1$, with nontrivial groups only in specific degrees.
• The cohomology subalgebra of elements of infinite order in $H^*(M; \mathbb{Z})$ is isomorphic to $H^*(S^4 \times S^4; \mathbb{Z})$, with Betti numbers $\beta_0 = \beta_8 = 1$, $\beta_4 = 2$, and $\beta_j = 0$ otherwise.
• The Euler characteristic of the source manifold $M$ is $\chi(M) = 4$, computed via $\chi(M) = \chi(\partial W_f) \chi(D^4) + \chi(W_f) \chi(S^3)$, with $\chi(\partial W_f) = 4$.
• The ranks of $H_j(M; \mathbb{Z})$ are zero for $j = 1,2,3$, and non-zero only for $j = 0,4,8$, with $\text{rank}(H_4(M; \mathbb{Z})) = 2$.
• Cup products in $H^*(M; \mathbb{Z})$ are trivial in all degrees, as shown by Theorem 5, despite nontrivial cohomology in degree 4 and 8.