Co-H-Structures on Moore Spaces of Type (G, 2)

Abstract

We consider the set (of homotopy classes) of co-H-structures on a Moore space M(G,n), where G is an abelian group and n is an integer ≥ 2. It is shown that for n > 2 the set has one element and for n = 2 the set is in one-one correspondence with Ext(G, G ⊗ G). We make a detailed investigation of the co-H-structures on M(G, 2) in the case G = ℤm, the integers mod m. We give a specific indexing of the co-H-structures on M(ℤm, 2) and of the homotopy classes of maps from M(ℤm, 2) to M(ℤn, 2) by means of certain standard homotopy elements. In terms of this indexing we completely determine the co-H-maps from M{ℤm, 2) to M(ℤn, 2) for each co-H-structure on M(ℤm, 2) and on M(ℤn, 2). This enables us to describe the action of the group of homotopy equivalences of M(ℤn, 2) on the set of co-H-structures of M(ℤm, 2). We prove that the action is transitive. From this it follows that if m is odd, all co-H-structures on M(ℤm, 2) are associative and commutative, and if m is even, all co-H-structures on M(ℤm, 2) are associative and non-commutative.

On certain co–H spaces related to Moore spaces

“…This co-H structure is unique [4]. Thus the composite with Σϕ is a co-H map, and the assertion follows from Corollary 3.2.…”

“…Hence it is a suspension, and so homotopy co-associative. If n ≥ 2 it is a double suspension, and the homotopy co-commutativity for n = 1 was proved in [4]. So suppose the assertion holds for k − 1, k ≥ 1.…”

On co-H-maps to the suspension of the projective plane

“…Let RP b a = RP b /RP a−1 and let X n be the nconnected cover of a space X. According to [20], the homotopy fibre of the inclusion P 3 (2) ⊂ ✲ BSO(3) is ΣRP 4 2 ∨ P 6 (2) and so there is a fibre sequence SO(3) ✲ ΣRP 4 2 ∨ P 6 (2) ✲ P 3 (2). It follows that there is a fibre sequence Ω(P 3 (2) 2 ) ∂ ✲ S 3 ✲ ΣRP 4 2 ∨ P 6 (2) ✲ P 3 (2) 2 , where the map S 3 → ΣRP 4 2 ∨ P 6 (2) is of degree 4 into the bottom cell of target space.…”

“…Assertion (1) follows from some standard arguments in homotopy theory. We introduce "combinatorial calculations" for the Hopf map to prove assertion (2). These combinatorial methods were introduced by Fred Cohen in [6] to attack the Barratt conjecture and have been applied to the James-Hopf maps [21].…”

Suspensions of crossed and quadratic complexes, co-H-structures and applications

“…The results of Section 5 are applied here to compute an explicit formula of the element c A from a free resolution of A (Theorem 7.5). In this way we show that if A is finitely generated, then c A is trivial if and only if A has an element of order 2 (Corollary 7.6), generalizing a result of Arkowitz and Golasiński ( [2]) for cyclic groups.…”