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The loop group and the cobar construction

Kathryn Hess, Andrew Tonks

Abstract

We prove that for any 1-reduced simplicial set X, Adams' cobar construction, ΩCX, on the normalised chain complex of X is naturally a strong deformation retract of the normalised chains CGX on the Kan loop group GX, opening up the possibility of applying the tools of homological algebra to transfering perturbations of algebraic structure from the latter to the former. In order to prove our theorem, we extend the definition of the cobar construction and actually obtain the existence of such a strong deformation retract for all 0-reduced simplicial sets.

The loop group and the cobar construction

Abstract

We prove that for any 1-reduced simplicial set X, Adams' cobar construction, ΩCX, on the normalised chain complex of X is naturally a strong deformation retract of the normalised chains CGX on the Kan loop group GX, opening up the possibility of applying the tools of homological algebra to transfering perturbations of algebraic structure from the latter to the former. In order to prove our theorem, we extend the definition of the cobar construction and actually obtain the existence of such a strong deformation retract for all 0-reduced simplicial sets.

Paper Structure

This paper contains 9 sections, 11 theorems, 89 equations.

Table of Contents

  1. Preliminaries
  2. Simplicial notions and notation
  3. The cobar construction
  4. The Kan loop group
  5. The Szczarba map
  6. The retraction map
  7. Definition of the map
  8. Properties of the retraction map
  9. Deformation retraction of the loop group

Key Result

Proposition 4

Let $X$ be a $0$-reduced simplicial set and $GX$ its Kan loop group. Then there is an isomorphism of rings \xymatrix{(\hat{\Omega} C X)_0\rto<1.1ex>_-\cong^-{\phi_{0}}&(CGX)_0\lto<1.1ex>^-{\psi_{0}} }determined by

Theorems & Definitions (28)

  • Remark 1
  • Remark 2
  • Remark 3
  • Proposition 4
  • proof
  • Definition 5
  • Lemma 6
  • Theorem 7
  • proof
  • Lemma 8
  • ...and 18 more