Jc. Thomas

The Category of a Map and the Grade of a Module

AbstractLetf: Y → X be a continuous map between connectedCW complexes. The homologyH*(F) of the homotopy fibre is then a module over the loop space homologyH* (ΩX). Theorem:If H*(F; R) and H*(ΩX; R) are R-free (R a principal ideal domain) then for some H*(ΩX; R)-projective moduleP=P>0and for some m ≤ cat f:

Free loop spaces of finite complexes have infinite category

Ext_{H_ * (\Omega X)}^m (H_ * (F);P) \ne 0.

Torsion in Loop Space Homology

Some applications are also given.

The Module of Holonomy of a Fibration

Abstract Let F → i E → p B be a fibration and let f : W → E be such that pf is homotopically trivial. Then p naturally extends to a fibration F′ → i′ E′ → p′ B with E ′≅ C f the cone of f . In our first result, we study the natural homomorphism of H ∗ (Ω B;R )-modules H ∗ ( F;R )→ H ∗ ( F ′; R ). When R = Q , we apply this result to study the cone length of a space, the free Lie ideals and a cellular construction of a graded Lie algebra.

Presentations of Algebras and the Whitehead Gamma-functor for Chain Algebras

Let X be a 1-connected space such that each H(j)(X;Z) is finitely generated. In this paper we prove that if the reduced homology of X with coefficients in a field is nonzero, then the Lusternik-Schnirelmann category of the free loop space is infinite.

Hopf-algebras and a Counterexample To a Conjecture of Anick

Let k be a field of positive characteristic and X be a simply connected space of the homotopy type of a finite type CW complex. The Postnikov fibre X([n]) of X is defined as the homotopy fibre of the n-equivalence fn: X --> X(n) coming from the Postnikov tower {X(n)} of X. We prove that if the Lusternik-Schnirelmann category of X is finite, then H*(X[n];k) contains a free module on a subalgebra K of H* (OMEGAX(n);k) such that H* (OMEGAX(n);k) is a finite-dimensional free K-module.