[Paper Review] Generalizations of Reeb spaces of special generic maps and applications to a problem of lifts of smooth maps

Naoki Kitazawa|arXiv (Cornell University)|May 20, 2018

Topological and Geometric Data Analysis3 references4 citations

TL;DR

This paper introduces generalized quotient maps onto Reeb spaces of special generic maps, extending them to maps into polyhedra and compact manifolds. It applies these generalizations to solve the problem of lifting Morse functions via embeddings into higher-dimensional Euclidean spaces, showing that any spherical Morse function on a closed manifold of dimension >2 can be lifted to an embedding into R^n with n ≥ max{(3m+3)/2, m+3}+1, such that composition with the canonical projection recovers the original function.

ABSTRACT

A Reeb space is defined as the space of all the connected components of inverse images of a smooth map, which is a fundamental tool in studying smooth manifolds using generic smooth maps whose codimensions are not positive such as Morse functions, their higher dimensional versions including fold maps and general stable maps. A special generic map is a fold map and a generalization of Morse functions with just 2 singular points on homotopy spheres and the Reeb space is a compact manifold whose dimension is equal to that of the target manifold and which can be immersed into the target manifold. In this paper, we generalize a quotient map onto a Reeb space of a special generic map. We define a map onto a polyhedron locally a quotient map induced from a special generic map. Moreover, we take advantage of the generalized maps to construct lifts of Morse functions of a certain class; the composition of the lift and the canonical projection is the original funciton. It is an answer of an explicit problem in the studies of lifts of smooth maps, or maps such that the compositions of the found maps and the canonical projections are original maps, which are fundamental and important in the studies of smooth maps and applications to algebraic and differential topology of manifolds.

Motivation & Objective

• To generalize quotient maps onto Reeb spaces of special generic maps to maps into polyhedra and compact manifolds.
• To address the open problem of lifting Morse functions to embeddings or immersions in higher-dimensional Euclidean spaces.
• To provide explicit constructions of lifts such that the composition with the canonical projection recovers the original smooth map.
• To extend previous results on lifts of special generic maps and pseudo special generic maps to higher codimensions.
• To establish conditions under which such lifts exist, particularly for non-orientable manifolds and higher-dimensional targets.

Proposed method

• Define a pseudo special generic map as a generalization of the Reeb space quotient map, locally modeled on special generic maps.
• Use the structure of the Reeb space and its collar neighborhood to extend local lifts to global embeddings.
• Apply techniques from [15] involving the simple connectivity of embedding spaces of spheres to extend trivial pseudo special generic maps.
• Construct an embedding lift into R^n by extending the map over the boundary collar of the Reeb space using trivial normal bundles.
• Ensure the composition of the embedding with the canonical projection π_{n,1} recovers the original Morse function.
• Utilize the fact that the space of smooth embeddings of S^{m-2} into R^{n-3} is simply connected to extend the lift globally.

Experimental results

Research questions

• RQ1 Can the Reeb space construction for special generic maps be generalized to maps into compact manifolds and polyhedra?
• RQ2 Under what conditions does a spherical Morse function on a closed manifold of dimension >2 admit a lift to an embedding in R^n such that composition with π_{n,1} recovers the original function?
• RQ3 What is the minimal dimension n of the target Euclidean space for which such a lift exists?
• RQ4 How do orientability and the topology of the source manifold affect the existence of such lifts?
• RQ5 Can the lift be constructed so that the composition with π_{n,3} yields a map equivalent to a trivial pseudo special generic map into R^3?

Key findings

• A spherical Morse function on a closed manifold of dimension m > 2 can be represented as the composition of a smooth embedding into R^n with the canonical projection π_{n,1}, where n ≥ max{(3m+3)/2, m+3}+1.
• The embedding lift can be constructed so that the composition with π_{n,3} yields a map equivalent to a trivial pseudo special generic map into a 2-dimensional compact manifold embedded in R^3.
• The construction relies on extending a local lift over the boundary collar of the Reeb space using the simple connectivity of embedding spaces of spheres.
• For non-orientable source manifolds, a lift as a special generic map into R^n with n > 2 cannot be constructed if the underlying pseudo special generic map is trivial.
• The result generalizes earlier work in [11], which required orientability and a slightly smaller n, by extending the construction to non-orientable cases and higher codimensions.