The homotopy groups of a homotopy group completion

DOI10.1007/S11856-019-1914-2zbMATH Open1461.19002arXiv1807.02613OpenAlexW2969651692MaRDI QIDQ2279928

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Publication date: 17 December 2019

Published in: (Search for Journal in Brave)

Abstract: Let $M$ be a topological monoid with homotopy group completion $OmegaBM$. Under a strong homotopy commutativity hypothesis on $M$, we show that $p{i}_k(OmegaBM)$ is the quotient of the monoid of free homotopy classes $[S^k,M]$ by its submonoid of nullhomotopic maps. We give two applications. First, this result gives a concrete description of the Lawson homology of a complex projective variety in terms of point-wise addition of spherical families of effective algebraic cycles. Second, we apply this result to monoids built from the unitary, or general linear, representation spaces of discrete groups, leading to results about lifting continuous families of characters to continuous families of representations.

Full work available at URL: https://arxiv.org/abs/1807.02613

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