Related papers: Constructing fold maps by surgery operations and t…

How can we design neural networks that allow for stable universal approximation of maps between topologically interesting manifolds? The answer is with a coordinate projection. Neural networks based on topological data analysis (TDA) use…

Here we shall consider the topology and dynamics associated to a wide class of matchbox manifolds, including a large selection of tiling spaces and all minimal matchbox manifolds of dimension one. For such spaces we introduce topological…

We give a survey of geometric approaches to the topological 4-dimensional surgery and 5-dimensional s-cobordism conjectures, with a focus on the study of surfaces in 4-manifolds. The geometric lemma underlying these conjectures is a…

Given a toric degeneration (a degeneration to a toric variety), over the complex numbers, we construct a surjective continuous map from a general fiber to the special fiber of the degeneration in the classical topology. The construction is…

Let M be a closed, oriented, n-dimensional manifold. In this paper we give a Morse theoretic description of the string topology operations introduced by Chas and Sullivan, and extended by the first author, Jones, Godin, and others. We do…

The Torelli group of a manifold is the group of all diffeomorphisms which act as the identity on the homology of the manifold. In this paper, we calculate the invariant part (invariant under the action of the automorphisms of the homology)…

Recent advancements in the discipline of quantum algorithms have displayed the importance of the geometry of quantum operators. Given this thrust, this paper develops a rigorous geometric framework to analyze how the Riemannian structure of…

In this paper we study the topology of the spaces Hol(M,P{n},k) of (basepoint preserving) holomorphic maps of a given degree k from a Riemann surface M of genus g>0 into the n-th complex projective space P{n}, n>0. Using symmetric products…

We calculate the RT-invariants of all oriented Seifert manifolds directly from surgery presentations. We work in the general framework of an arbitrary modular category as in [V. G. Turaev, Quantum invariants of knots and 3--manifolds, de…

In this paper we prove the existence of a natural mapping from the surgery exact sequence for topological manifolds to the analytic surgery exact sequence of N. Higson and J. Roe. This generalizes the fundamental result of Higson and Roe,…

We give a description of degree-one maps between closed, oriented 3-manifolds in terms of surgery. Namely, we show that there is a degree-one map from a closed, oriented 3-manifold $M$ to a closed, oriented 3-manifold $N$ if and only if $M$…

Our main result is a generalization of Cappell's 5-dimensional splitting theorem. As an application, we analyze, up to internal s-cobordism, the smoothable splitting and fibering problems for certain 5-manifolds mapping to the circle. For…

This paper describes how to recover the topology of a closed manifold $M$ from a good Morse function $f$ on $M$. The essential method was suggested by Cohen, Jones and Segal. They constructed a topological category $C_{f}$ and claimed that…

For a connected locally path-connected topological space $X$ and a continuous function $f$ on it such that its Reeb graph $R_f$ is a finite topological graph, we show that the cycle rank of $R_f$, i.e., the first Betti number $b_1(R_f)$, in…

Manifold learning is a popular and quickly-growing subfield of machine learning based on the assumption that one's observed data lie on a low-dimensional manifold embedded in a higher-dimensional space. This thesis presents a mathematical…

We study {\em $\nabla$-Sobolev spaces} and {\em $\nabla$-differential operators} with coefficients in general Hermitian vector bundles on Riemannian manifolds, stressing a coordinate free approach that uses connections (which are typically…

Riemannian Geometry, Topology and Dynamics permit to introduce partially defined holomorphic functions on the variety of representations of the fundamental group of a manifold. The functions we consider are the complex valued Ray-Singer…

Submersions with definite folds are submersions on manifolds with boundary whose restrictions to the boundary are definite fold maps. In this paper, we study the properties from the viewpoint of differential topology of manifolds with…

Intermediate feature representations represent the backbone for the expressivity and adaptability of deep neural networks. However, their geometric structure remains poorly understood. In this submission, we provide indirect insights into…

This note presents a procedure of constructing a higher dimensional sphere map from a lower dimensional one and gives an explicit formula for smooth sphere map with a given degree. As an application a new proof of a generalized…