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Foundations of geometric cohomology: from co-orientations to product structures

Greg Friedman, Anibal M. Medina-Mardones, Dev Sinha

Abstract

This manuscript develops a geometric approach to ordinary cohomology of smooth manifolds, constructing a cochain complex model based on co-oriented smooth maps from manifolds with corners. Special attention is given to the pull-back product of such smooth maps, which provides our geometric cochains with a partially defined product structure inducing the cup product in cohomology. A parallel treatment of homology is also given allowing for a geometric unification of the contravariant and covariant theories.

Foundations of geometric cohomology: from co-orientations to product structures

Abstract

This manuscript develops a geometric approach to ordinary cohomology of smooth manifolds, constructing a cochain complex model based on co-oriented smooth maps from manifolds with corners. Special attention is given to the pull-back product of such smooth maps, which provides our geometric cochains with a partially defined product structure inducing the cup product in cohomology. A parallel treatment of homology is also given allowing for a geometric unification of the contravariant and covariant theories.
Paper Structure (82 sections, 182 theorems, 294 equations, 5 figures)

This paper contains 82 sections, 182 theorems, 294 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Manifolds with corners
  3. Boundaries
  4. Transversality
  5. Achieving transversality
  6. Pullbacks and fiber products
  7. Some further properties of transversality and fiber products
  8. Fiber products with more than two inputs
  9. Orientations and co-orientations
  10. Orientations
  11. Orientations of fiber products
  12. Fiber products of immersions
  13. Co-orientations
  14. Normal co-orientations of immersions and co-orientations of boundaries
  15. Quillen co-orientations
  16. ...and 67 more sections

Key Result

Lemma 2.15

Let $f \colon V \to M$ and $g \colon W \to M$ be maps from manifolds with corners to a manifold without boundary. Then $f$ and $g$ are transverse if and only if $fi_{\partial^j} \colon \partial^jV \to M$ and $gi_{\partial^k} \colon \partial^kW \to M$ are naively transverse for all $j,k$.

Figures (5)

  • Figure 1:
  • Figure 2:
  • Figure 3:
  • Figure 4:
  • Figure 5: The first cellular decomposition of a torus pictured above does not represent the geometric realization of a cubical complex, as each square has the same set of vertices. On the right, each square can be coherently identified with the standard square with initial vertex in the lower left corner and final vertex in the upper right corner. Therefore, (B) depicts a cubical structure on the torus.

Theorems & Definitions (443)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Remark 2.4
  • Definition 2.6
  • Example 2.7
  • Definition 2.8
  • Definition 2.9
  • Remark 2.10
  • Example 2.11
  • ...and 433 more