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EQUIVARIANT CALCULUS OF FUNCTORS AND$\mathbb{Z}/2$-ANALYTICITY OF REAL ALGEBRAIC $K$-THEORY

Part of: $K$-theory of forms Homotopy theory

Published online by Cambridge University Press:  01 April 2015

Emanuele Dotto
Emanuele Dotto*
Affiliation:
Mathematics, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Building E18, Cambridge, 02139-4307, USA (dotto@mit.edu)
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Abstract

We define a theory of Goodwillie calculus for enriched functors from finite pointed simplicial $G$-sets to symmetric $G$-spectra, where $G$ is a finite group. We extend a notion of $G$-linearity suggested by Blumberg to define stably excisive and ${\it\rho}$-analytic homotopy functors, as well as a $G$-differential, in this equivariant context. A main result of the paper is that analytic functors with trivial derivatives send highly connected $G$-maps to $G$-equivalences. It is analogous to the classical result of Goodwillie that ‘functors with zero derivative are locally constant’. As the main example, we show that Hesselholt and Madsen’s Real algebraic $K$-theory of a split square zero extension of Wall antistructures defines an analytic functor in the $\mathbb{Z}/2$-equivariant setting. We further show that the equivariant derivative of this Real $K$-theory functor is $\mathbb{Z}/2$-equivalent to Real MacLane homology.

Keywords

Hermitian K-theoryequivariant homotopy theory

MSC classification

Primary: 19G38: Hermitian $K$-theory, relations with $K$-theory of rings 55P91: Equivariant homotopy theory
Type
Research Article
Information
Journal of the Institute of Mathematics of Jussieu , Volume 15 , Issue 4 , October 2016 , pp. 829 - 883
Copyright
© Cambridge University Press 2015 

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