Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

The stable cohomology of self-equivalences of connected sums of products of spheres

Robin Stoll

Abstract

We identify the cohomology of the stable classifying space of homotopy automorphisms (relative to an embedded disk) of connected sums of $S^k \times S^l$, where $3 \le k < l \le 2k - 2$. The result is expressed in terms of Lie graph complex homology.

The stable cohomology of self-equivalences of connected sums of products of spheres

Abstract

We identify the cohomology of the stable classifying space of homotopy automorphisms (relative to an embedded disk) of connected sums of , where . The result is expressed in terms of Lie graph complex homology.
Paper Structure (24 sections, 32 theorems, 222 equations)

This paper contains 24 sections, 32 theorems, 222 equations.

Table of Contents

  1. Introduction
  2. Sketch of the proof of \ref{['thm:intro_main']}.
  3. Related work and further research.
  4. Structure of this paper.
  5. Acknowledgments.
  6. Preliminaries
  7. Graded vector spaces
  8. Double complexes
  9. Group actions
  10. Kan extensions
  11. Symmetric sequences
  12. Schur functors
  13. Differential graded Lie algebras
  14. The convolution Lie algebra
  15. The directed graph complex
  16. ...and 9 more sections

Key Result

Theorem 1.1

Let $3 \le k < l \le 2k - 2$ and $2 \le g$ be integers. Then there is, in cohomological degrees $\le g - 2$, an isomorphism of graded algebras compatible with the stabilization maps on the left hand side. Here we set ${\operatorname{GL}_{}}({\mathbb{Z}}) \coloneqq \mathinner{\operatorname{colim} \displaylimits _{g \in {\mathbb{N}}}} {\operatorname{GL}_{g}}({\mathbb{Z}})$, denote by ${\mathfrak{UG

Theorems & Definitions (135)

  • Theorem 1.1: see \ref{['thm:main']}
  • Remark
  • Remark
  • Corollary 1.2
  • Theorem 1.3: see \ref{['lemma:trunc_dgc']}
  • Remark
  • Theorem 1.4: see \ref{['lemma:UGC_diff']}
  • Remark
  • Remark
  • Remark 2.7
  • ...and 125 more