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Infinite Loop Spaces, Dyer–Lashof Algebra, Cohomology of the Infinite Symmetric Group and Modular Invariants

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Algebraic Modeling of Topological and Computational Structures and Applications (AlModTopCom 2015)

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 219))

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Abstract

In this lecture note we survey results obtained under the research program Thalis (Kechagias, J. Homotopy Relat. Struct. 8(2), 201–229, (2013), [24], J. Pure Appl. Algebra 219(4), 839–863, (2015), [25]) and place them in the context of algebraic topology. It is divided into two parts. In the first part, we survey infinite loop spaces, \(\varOmega \)-spectra, and their relation with the symmetric groups. In the second part, we express the component Dyer–Lashof coalgebras as subalgebras of a cofree unstable coalgebra on two cogenerators using an extension of the Peterson conjecture. We also compare and approximate \(H^{*}\left( Q_{0}S^{0};\mathbb {Z}/p \mathbb {Z}\right) \) with certain free objects using modular invariants. A new basis for \(H^{*}\left( B\varSigma _{\infty }; \mathbb {Z}/p \mathbb {Z}\right) \) is provided.

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Authors and Affiliations

  1. Department of Mathematics, University of Ioannina, 45500, Ioannina, Greece

    Nondas E. Kechagias

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  1. Nondas E. Kechagias
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Correspondence to Nondas E. Kechagias .

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Editors and Affiliations

  1. Department of Applied Mathematics, National Technical University of Athens, Athens, Greece

    Sofia Lambropoulou

  2. School of Chemical Engineering, National Technical University of Athens, Athens, Greece

    Doros Theodorou

  3. Department of Applied Mathematics, National Technical University of Athens, Athens, Greece

    Petros Stefaneas

  4. Department of Mathematics, University of Illinois at Chicago, Chicago, Illinois, USA

    Louis H. Kauffman

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Kechagias, N.E. (2017). Infinite Loop Spaces, Dyer–Lashof Algebra, Cohomology of the Infinite Symmetric Group and Modular Invariants. In: Lambropoulou, S., Theodorou, D., Stefaneas, P., Kauffman, L. (eds) Algebraic Modeling of Topological and Computational Structures and Applications. AlModTopCom 2015. Springer Proceedings in Mathematics & Statistics, vol 219. Springer, Cham. https://doi.org/10.1007/978-3-319-68103-0_10

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