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$\mathbb{Z}_k$-stratifolds

Andrés Angel, Arley Fernando Torres, Carlos Segovia

Abstract

Generalizing the ideas of $\mathbb{Z}_k$-manifolds from Sullivan and stratifolds from Kreck, we define $\mathbb{Z}_k$-stratifolds. We show that the bordism theory of $\mathbb{Z}_k$-stratifolds is sufficient to represent all homology classes of a $CW$-complex with coefficients in $\mathbb{Z}_k$. We present a geometric interpretation of the Bockstein long exact sequences and the Atiyah-Hirzebruch spectral sequence for $\mathbb{Z}_k$-bordism ($k$ an odd number). Finally, for $p$ an odd prime, we give geometric representatives of all classes in $H_*(B\mathbb{Z}_p;\mathbb{Z}_p)$ using $\mathbb{Z}_p$-stratifolds.

$\mathbb{Z}_k$-stratifolds

Abstract

Generalizing the ideas of -manifolds from Sullivan and stratifolds from Kreck, we define -stratifolds. We show that the bordism theory of -stratifolds is sufficient to represent all homology classes of a -complex with coefficients in . We present a geometric interpretation of the Bockstein long exact sequences and the Atiyah-Hirzebruch spectral sequence for -bordism ( an odd number). Finally, for an odd prime, we give geometric representatives of all classes in using -stratifolds.

Paper Structure

This paper contains 13 sections, 32 theorems, 59 equations, 9 figures.

Table of Contents

  1. Introduction
  2. Acknowledgements
  3. $\mathbb{Z}_k$-Manifolds
  4. Stratifolds
  5. ${\mathbb Z}_k$-Stratifolds
  6. The Bockstein sequence
  7. Fundamental classes
  8. A geometric description of the Atiyah--Hirzebruch spectral sequence for $\mathbb{Z}_k$-coefficients
  9. Geometric representatives of non-representable classes
  10. $\mathbb{Z}_2$-stratifold homology is stratifold homology with $\mathbb{Z}_2$-coefficients
  11. Appendix
  12. Regular values for $\mathbb{Z}_k$-stratifolds
  13. Mayer--Vietoris sequence

Key Result

Theorem 1.1

An isomorphism exists between $\mathbb{Z}_k$-stratifold homology theory and singular homology with $\mathbb{Z}_k$-coefficients. This isomorphism is valid for all CW-complexes and is compatible with the Bockstein homomorphisms.

Figures (9)

  • Figure 1: Representation of the Klein bottle as the quotient space of a $\mathbb{Z}_2$-manifold.
  • Figure 2: A closed $\mathbb{Z}_3$-manifold.
  • Figure 3: A $\mathbb{Z}_3$-manifold with boundary.
  • Figure 4: Left: a $\mathbb{Z}_3$-manifold with boundary / Right: the boundary $\partial B$ after quotient.
  • Figure 5: A closed $\mathbb{Z}_3$-stratifold.
  • ...and 4 more figures

Theorems & Definitions (89)

  • Theorem 1.1
  • Theorem 1.2
  • Definition 2.2
  • Definition 2.3
  • Example 2.4
  • Example 2.5
  • Example 2.6
  • Definition 2.7
  • Definition 2.8
  • Example 2.9
  • ...and 79 more