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Quantitative PL bordism

Fedor Manin, Bena Tshishiku, Shmuel Weinberger

Abstract

We study PL bordism theories from a quantitative perspective. Such theories include those of PL manifolds, ordinary homology theory, as well as various more exotic theories such as bordism of Witt spaces. In all these cases we show that a null-bordant cycle of bounded geometry and $V$ simplices has a filling of bounded geometry whose number of simplices is slightly superlinear in $V$. This bound is similar to that found in our previous work on smooth cobordism.

Quantitative PL bordism

Abstract

We study PL bordism theories from a quantitative perspective. Such theories include those of PL manifolds, ordinary homology theory, as well as various more exotic theories such as bordism of Witt spaces. In all these cases we show that a null-bordant cycle of bounded geometry and simplices has a filling of bounded geometry whose number of simplices is slightly superlinear in . This bound is similar to that found in our previous work on smooth cobordism.
Paper Structure (26 sections, 37 theorems, 120 equations, 1 figure, 2 tables)

This paper contains 26 sections, 37 theorems, 120 equations, 1 figure, 2 tables.

Table of Contents

  1. Introduction
  2. Discussion
  3. Overview of proof
  4. Notation
  5. Examples and corollaries
  6. PL manifolds
  7. General pseudomanifolds
  8. Other classes
  9. Relation to secondary invariants
  10. Efficient Whitney embeddings
  11. Simplicial Pontrjagin--Thom constructions
  12. Classes of singularities
  13. Thom spaces
  14. Modifications for bordism homology
  15. The Thom map and its nullhomotopy
  16. ...and 11 more sections

Key Result

Theorem 1.1

Let $M$ be a PL $k$-manifold with bounded geometry of type $L$ and $V$ top-dimensional simplices. Then if $M$ is a PL boundary, it bounds a manifold $W$ with bounded geometry of type $L'$ (depending on $k$ and $L$) so that the number of simplices in $W$ is bounded by a function (again depending inex

Figures (1)

  • Figure 1: Schematic illustrations of $f(cP)$, $N$ for a typical triple $(P,f,N)$.

Theorems & Definitions (82)

  • Theorem 1.1
  • Theorem : CoTh
  • Corollary 2.1
  • proof
  • Theorem 2.2
  • proof
  • Lemma 2.3
  • proof
  • Corollary 2.4
  • proof
  • ...and 72 more