The first k-invariant of a double loop space is trivial
Let X be a connected simple CW-complex and n an integer \(\geq 2\) such that \(\pi_ iX=0\) for \(1<i<n\); the first k-invariant of X is then \(k^{n+1}[X]\in H^{n+1}(k(\pi_ 1X,1);\pi_ nX)\). The main result of this paper is the following theorem: let p be a prime number and assume that X is a connected (2p-2)-fold loop space satisfying \(\pi_ iX=0\) for \(1<i<(2p-2)\), then its first k-invariant fulfills \(L_ pk^{2p- 1}[X]=0\), where \(L_ p\) denotes the product of all primes \(q<p\). In particular when \(p=2\), the theorem asserts that the first k-invariant of a connected double loop space X is trivial: \(k^ 3[X]=0\) in \(H^ 3(K(\pi_ 1X,1);\pi_ 2X)\). Two different proofs are given: the first one is based on the study of the (co)homology suspension for Eilenberg- MacLane spaces, and the second is obtained from known results on classical cohomology operations.
first k-invariant of a connected double loop space
(co)homology suspension for Eilenberg- MacLane spaces
scientific article; zbMATH DE number 4100177