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The Pontrjagin Dual of 3-Dimensional Spin Bordism

Greg Brumfiel, John Morgan

Abstract

For each space X we define an explicit group, G(X), functorially in X. This group is constructed from the groups of cochains on X. Furthermore, we construct an explicit functorial pairing with values in R/Z between the cochain representatives for elements of G(X) and maps of closed 3-dimensional spin manifolds to X. This pairing induces a pairing between G(X) and the 3-dimensional spin bordism group of X and identifies each with the Pontrjagin dual of the other.

The Pontrjagin Dual of 3-Dimensional Spin Bordism

Abstract

For each space X we define an explicit group, G(X), functorially in X. This group is constructed from the groups of cochains on X. Furthermore, we construct an explicit functorial pairing with values in R/Z between the cochain representatives for elements of G(X) and maps of closed 3-dimensional spin manifolds to X. This pairing induces a pairing between G(X) and the 3-dimensional spin bordism group of X and identifies each with the Pontrjagin dual of the other.

Paper Structure

This paper contains 39 sections, 39 theorems, 193 equations.

Table of Contents

  1. Introduction
  2. Background and Statement of the Main Result
  3. Remarks on Compact Topologies
  4. Comparison with Homotopy Theoretic Approaches
  5. The Atiyah-Hirzebruch Spectral Sequence and Filtrations
  6. The Spectral Sequence and Filtration of $\Omega_3^{spin}(X)$
  7. The Filtration of $G(X)$
  8. Comparison of the Filtrations
  9. Construction of the Group G(X)
  10. The Definition of the Product on Triples of Cochains
  11. Natural Operations From Cochains to Cocycles
  12. The Definition of the Relations on Triples of Cochains
  13. Discussion of Associativity and Inverses
  14. Discussion of Commutators and Commutativity
  15. The Definition of the Group $G = G(X)$
  16. ...and 24 more sections

Key Result

Theorem 1.1

Fix a space $X$. For any abelian group $M$ let $C^*(X; M)$ and $Z^*(X; M)$ denote the singular cochains and singular cocycles with values in $M$. Let with its natural compact topology. (See Section sect1.2.) We define Then $G$ is a functor from the homotopy category to the category of compact abelian groups. (See Corollary homfunct.)

Theorems & Definitions (101)

  • Theorem 1.1
  • Theorem 1.2
  • Proposition 2.1
  • Theorem 2.2
  • Definition 3.1
  • Proposition 3.2
  • proof
  • Remark 3.3
  • Remark 3.4
  • Lemma 3.5
  • ...and 91 more