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The homotopy groups of a homotopy group completion

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Abstract

Let M be a topological monoid with homotopy group completion ΩBM. Under a strong homotopy commutativity hypothesis on M, we show that πk(ΩBM) is the quotient of the monoid of free homotopy classes [Sk, 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 pointwise 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.

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Authors and Affiliations

  1. Departmenet of Mathematical Sciences, Indiana University-Purdue University Indianapolis, 402 N. Blackford, LD 270, Indianapolis, IN, 46202, USA

    Daniel A. Ramras

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  1. Daniel A. Ramras
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Correspondence to Daniel A. Ramras.

Additional information

The author was partially supported by the Simons Foundation (#279007).

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Ramras, D.A. The homotopy groups of a homotopy group completion. Isr. J. Math. 234, 81–124 (2019). https://doi.org/10.1007/s11856-019-1914-2

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  • DOI: https://doi.org/10.1007/s11856-019-1914-2

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