The homotopy groups of a homotopy group completion

Let M be a topological monoid with homotopy group completion Ω BM . Under a strong homotopy commutativity hypothesis on M , we show that π k (Ω BM ) 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...

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Bibliographic Details
Published inIsrael journal of mathematics Vol. 234; no. 1; pp. 81 - 124
Main Author Ramras, Daniel A.
Format Journal Article
LanguageEnglish
Published Jerusalem The Hebrew University Magnes Press 01.10.2019
Springer Nature B.V
Subjects
Algebra
Analysis
Applications of Mathematics
Commutativity
Group theory
Group Theory and Generalizations
Homology
Mathematical and Computational Physics
Mathematics
Mathematics and Statistics
Monoids
Quotients
Representations
Theoretical
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Summary:Let M be a topological monoid with homotopy group completion Ω BM . Under a strong homotopy commutativity hypothesis on M , we show that π k (Ω BM ) 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 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.
ISSN:0021-2172
1565-8511
DOI:10.1007/s11856-019-1914-2