The homotopy groups of a homotopy group completion

2018/07/07 by Daniel A. Ramras, Ramras, Daniel A.
Mathematics · #14D20 #14F43 (secondary) #55R35 (primary) #Algebraic Topology (math.AT) #FOS: Mathematics #math.AT #msc:14D20 #msc:14F43 #msc:55R35

paper · pdf · doi:10.48550/arxiv.1807.02613

40 pages. The results in this paper replace Section 2-4 of arXiv:1607.06430. v2: Various small corrections and clarifications. Accepted for publication in Israel J. Math. (v2 contains two appendices that will not appear in the journal version)

arxiv created 2018/12/30 · arxiv updated 2019/01/01

Abstract

Let M be a topological monoid with homotopy group completion ΩBM. Under a strong homotopy commutativity hypothesis on M, we show that π_k (ΩBM) 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.

Citations

• The topological Atiyah-Segal map

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