Torsion in loop space homology of rationally contractible spaces (Q5961639)

The aim of this paper is the study of the growth of torsion, in loop space homology under finiteness conditions on the space. Finiteness properties considered here are of different nature, either the space itself admits a homology exponent or else the primitive subspace of the homology is finite dimensional. Interesting remarks conclude the paper. To give a flavor of this paper we just quote the two first results.NEWLINENEWLINENEWLINETheorem 1: Let \(C\) be a \(c\)-connected differential graded coalgebra, \(c\geq 1\) and \(\Omega C\) be the cobar construction. Suppose \(C\) admits a homology exponent \(r\). Let \(d\) and \(e\) be positive integers such that \(e\geq (d-c+1)/c\). Then \(r^e\) annihilates \(H_i(\Omega X)\) for each \(1\leq i\leq d\).NEWLINENEWLINENEWLINEThe approximation given by Theorem 1 is best possible. The least \(d\), for which \(r^{e-1}\) might fail to be an exponent for \(H_d(\Omega C)\) is \(d=e(2k-1)\).NEWLINENEWLINENEWLINETheorem 2: For each pair of positive integers \(t\) and \(k\), there exists a \((2k-1)\)-connected differential graded coalgebra \(C^{k,t}\) with a homology exponent \(t\), such that for every positive integer \(e\), \(H_{e(2k-1)}(\Omega C^{k,1})\) contains an element of order \(t^e\).NEWLINENEWLINENEWLINEAn application of these purely algebraic results is obtained by considering the classifying space \(BG\), where \(G\) is a finite group.