The second stable homotopy group of the Eilenberg–Maclane space

Abstract

We prove that for a finitely generated group G, the second stable homotopy

Finiteness of homotopy groups related to the non-abelian tensor product

“…Many authors have studied bounds on the order of π 3 (SK (G, 1)) (cf. [1,6,7,17]). We deduce a finiteness criterion for π 3 (S X) in terms of π 2 (X ) and the number of tensors T ⊗ (G), where π 1 (X ) ∼ = G and S X is the suspension of the space X (see Remark 2.4).…”

Finiteness of homotopy groups related to the non-abelian tensor product

“…Many authors had studied bounds to the order of π 3 (SK(G, 1)) (cf. [1,6,7,17]). Now, we can deduce a finiteness criterion for π 3 (SX) in terms of π 2 (X) and the number of tensors T ⊗ (G), where π 1 (X) ∼ = G and SX is the suspension of a space X (see Remark 2.4, below).…”

On the Exponent of the Schur Multiplier

“…To prove (1), it is enough to restrict ourselves to p groups using a standard argument given in Theorem 4, Chapter IX of [27]. A. Lubotzky and A. Mann showed that (1) holds for powerful p groups( [15]), M. R. Jones in [13] proved that (1) holds for groups of class 2, P. Moravec showed that (1) holds for groups of nilpotency class at most 3, odd order class 4, potent p groups, metabelian p groups of exponent p, p groups of class at most p − 2 ( [19], [20], [21]) and some other classes of groups. The general validity of (1) was disproved by A. J. Bayes, J. Kautsky and J. W. Wamsley in [3].…”

“…This problem has remained open even for finite p groups of class 5 having odd exponent. The purpose of this paper is to prove (1) for finite p groups of class 5 with odd exponent and to also prove the above mentioned results of [15], [19], [20] and [21] for odd primes and hence proving all these results using a common technique and bringing them under one umbrella. We briefly describe the organization of the paper by listing the main results according to their sections.…”