arXiv subjects
Publications and source records attributed to Filipp Levikov.
We prove a non-linear version of a theorem of Grayson which is an analogue of the Fundamental Theorem of Algebraic $K$-theory and identify the $K$-theory of the endomorphism category over a space $X$ in terms of reduced $K$-theory of a certain localisation of the category of $\NN$-spaces over $X$. In particular we generalise the result of Klein and Williams describing the nil-terms of $A$-theory in terms of $K$-theory of nilpotent endomorphisms.
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.