From $$({\mathbb {Z}},X)$$ ( Z , X ) -modules to homotopy cosheaves
| Autor: | Filipp Levikov |
|---|---|
| Rok vydání: | 2015 |
| Předmět: |
Pure mathematics
Algebra and Number Theory Functor Homotopy 010102 general mathematics Pontryagin class Mathematics::Algebraic Topology 01 natural sciences Surgery obstruction Chain (algebraic topology) Mathematics::K-Theory and Homology Surgery exact sequence Mathematics::Category Theory 0103 physical sciences 010307 mathematical physics Geometry and Topology 0101 mathematics Algebraic number Equivalence (measure theory) Mathematics |
| Zdroj: | Journal of Homotopy and Related Structures. 11:261-289 |
| ISSN: | 1512-2891 2193-8407 |
| DOI: | 10.1007/s40062-015-0105-z |
| Popis: | We construct a functor from the category of $(\mathbb{Z},X)$-modules of Ranicki (cf. \cite{Ra92}) to the category of homotopy cosheaves of chain complexes of Ranicki-Weiss (cf. \cite{RaWei10}) inducing an equivalence on $L$-theory. The $L$-theory of $(\mathbb{Z},X)$-modules is central in the algebraic formulation of the surgery exact sequence and in the construction of the total surgery obstruction by Ranicki, as described in \cite{Ra79}. The symmetric $L$-theory of homotopy cosheaf complexes is used by Ranicki-Weiss in \cite{RaWei10}, to reprove the topological invariance of rational Pontryagin classes. The work presented here may be considered as an addendum to the latter article and suggests some translation of ideas of Ranicki into the language of homotopy chain complexes of cosheaves. |
| Databáze: | OpenAIRE |
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