Simplicial and operad methods in algebraic topology

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In recent years, for solving problems of algebraic topology and, in particular, difficult problems of homotopy theory, algebraic structures more complicated than just a topological monoid, an algebra, a coalgebra, etc. A convenient language for describing various structures arising naturally on topological spaces and on their cohomology and homotopy groups is the language of operads and algebras over an operad.

This language was proposed by J. May in the s to describe the structures on various loop spaces. This book presents a detailed study of the concept of an operad in the categories of topological spaces and of chain complexes.

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The notions of an algebra and a coalgebra over an operad are introduced, and their properties are investigated. The algebraic structure of the singular chain complex of a topological space is explained, and it is shown how the problem of homotopy classification of topological spaces can be solved using this structure.


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For algebras and coalgebras over operads, standardconstructions are defined, particularly the bar and cobar consturctions. We construct a new model for genuine equivariant symmetric monoidal categories, based on applying the equivariant combinatorics of G-trees to the monad describing permutative categories. In particular, this provides a comparison and connection between all known equivariant generalizations of symmetric monoidal categories.

Moreover, there is an equivariant Barratt-Priddy-Quillen Theorem in this context, consistent with pre-existing versions. We show that the Blumberg-May-Hill candidate for realizing any indexing system as an N-infinity operad fails for general groups.

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We encode single-colored operads in G-spaces as a coefficient system of monoids in enriched functor categories, which in particular yield evaluations for any finite H-set. Moreover, this presentation explicitly presents the norm maps on N-infinity algebras. Research Interests: I am interested in topics in algebraic topology related to equivariant stable homotopy theory, in particular: equivariant operads, dendroidal sets, model categories, and structured ring spectra. On the homotopy theory of equivariant colored operads ArXiv preprint: Equivariant dendroidal Segal spaces and G-infinity-operads ArXiv preprint: Genuine equivariant operads ArXiv preprint: Power operations and transfers in equivariant cohomology theories in preparation [draft available upon request] with Bert Guillou and Nathaniel Stapleton Any ring spectrum E gives rise to a multiplicative cohomology theory equipped with power operations, and it is necessary to quotient out by a certain submodule in order to make these operations additive.

Equivariant dendroidal sets and simplicial operads in preparation [draft available upon request] with Luis Pereira We complete our survey of models for equivariant operads with norms up to homotopy, by showing that the Dwyer-Kan model structure on equivariant simplicial operads is Quillen equivalent to the G-infinity-operad model structure on dendroidal sets. A new model and comparison of genuine equivariant symmetric monoidal categories in preparation [draft available upon request] with Luis Pereira We construct a new model for genuine equivariant symmetric monoidal categories, based on applying the equivariant combinatorics of G-trees to the monad describing permutative categories.

Counterexamples to N-infinity realization candidates Available here updated 5 Jan We show that the Blumberg-May-Hill candidate for realizing any indexing system as an N-infinity operad fails for general groups. Presenting equivariant operads Available here updated 5 Jan We encode single-colored operads in G-spaces as a coefficient system of monoids in enriched functor categories, which in particular yield evaluations for any finite H-set. Dissertation My thesis, Models for Equivariant Operads can be found here. Equivariant power operations and the transfer University of Kentucky, Topology Seminar , January Final Price Rs.

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Apply Exchange. A convenient language for describing various structures arising naturally on topological spaces and on their cohomology and homotopy groups is the language of operads and algebras over an operad. This book presents a study of the concept of an operad in the categories of topological spaces and of chain complexes.

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