From \(({\mathbb {Z}},X)\) -modules to homotopy cosheaves

详细信息    查看全文

• 作者：Filipp Levikov
• 刊名：Journal of Homotopy and Related Structures
• 出版年：2016
• 出版时间：June 2016
• 年：2016
• 卷：11
• 期：2
• 页码：261-289
• 全文大小：748 KB
• 刊物主题：Algebraic Topology; Algebra; Functional Analysis; Number Theory;
• 出版者：Springer Berlin Heidelberg
• ISSN：1512-2891
• 卷排序：11

文摘

We construct a functor from the category of \((\mathbb {Z},X)\)-modules of Ranicki (Algebraic L-theory and topological nanifolds. Cambridge University Press, 1992) to the category of homotopy cosheaves of chain complexes of Ranicki and Weiss (Geom Dedic 148, 2010) 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 (Lect Notes Math 763:275–316 1979). The symmetric \(L\)-theory of homotopy cosheaf complexes is used by Ranicki and Weiss (Geom Dedic 148, 2010), 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 cosheaves of chain complexes.Keywords\(L\)-theoryRational Pontryagin classesHomotopy cosheavesTotal surgery obstruction