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Finite subset spaces of spheres

Jacob Mostovoy

Abstract

We find the complete rational homology for the finite subset spaces of a $d$-dimensional sphere. We also determine the integral homology in top $d$ degrees and obtain a partial description of it in codimension $d$.

Finite subset spaces of spheres

Abstract

We find the complete rational homology for the finite subset spaces of a -dimensional sphere. We also determine the integral homology in top degrees and obtain a partial description of it in codimension .
Paper Structure (11 sections, 16 theorems, 61 equations, 1 figure)

This paper contains 11 sections, 16 theorems, 61 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Three types of finite subset spaces
  3. Subset spaces for based polyhedra
  4. Finite subset spaces with unlimited number of points
  5. The case of $X$ of dimension $d$
  6. A torsion class in codimension $d$
  7. The cohomology of the configuration spaces $C_n(\mathbb{R}^d)$
  8. The homology of $\exp_n S^d$
  9. The spectral sequence for the homology of $\overline{\exp}_n S^d$
  10. The homology of $\overline{\exp}_n S^d$ in top $d$ degrees
  11. The homology of $\exp_n S^d$

Key Result

Theorem 1

The space $\exp_n S^d$ has the same rational homology as $S^{nd}\vee S^{(n-1)d}$ for $d$ even, and as $S^{[\frac{n+1}{2}](d+1)-1}$ for $d>1$ odd.

Figures (1)

  • Figure 1: The spectral sequence for the homology of $\overline{\exp}_n S^d$

Theorems & Definitions (19)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Lemma 4
  • Lemma 5
  • proof
  • Proposition 6
  • Proposition 7
  • proof
  • Theorem 8
  • ...and 9 more