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A factorization homology primer. (English) Zbl 1476.55020

Miller, Haynes (ed.), Handbook of homotopy theory. Boca Raton, FL: CRC Press. CRC Press/Chapman Hall Handb. Math. Ser., 39-101 (2020).
The Handbook of homotopy theory has been reviewed in [Zbl 1468.55001].
Summary: This chapter introduces factorization homology – or factorization algebras – in the topological setting. Factorization homology has three essential features making it technically advantageous: Local-to-global principle: \(\otimes\)-excision, generalizing the Eilenberg-Steenrod axioms, Filtration: a generalization of the Goodwillie-Weiss embedding calculus and Duality: Poincaré/Koszul duality. The chapter discusses a classification of sheaves on an \(\infty\)-category \(\mathcal{M}\mathsf{fld}_n\) of \(n\)-manifolds and embeddings among them: sheaves on \(\mathcal{M}\mathsf{fld}_n\) are \(n\)-dimensional tangential structures. It identifies values of factorization homology of \(\mathcal{D}\mathsf{isk}_n\)-algebras in spaces, with its Cartesian monoidal structure, as twisted compactly supported mapping spaces. The chapter describes filtrations and cofiltrations of factorization homology, whose layers are explicit in terms of configuration spaces. These (co)filtrations offer access to identifying and controlling factorization homology.
For the entire collection see [Zbl 1468.55001].

MSC:

55N40 Axioms for homology theory and uniqueness theorems in algebraic topology
57R56 Topological quantum field theories (aspects of differential topology)
57N35 Embeddings and immersions in topological manifolds
57R19 Algebraic topology on manifolds and differential topology
55-02 Research exposition (monographs, survey articles) pertaining to algebraic topology

Citations:

Zbl 1468.55001
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