Twisted ambidexterity in equivariant homotopy theory: Two approaches

Abstract

This dissertation explores the phenomenon of twisted ambidexterity in equivariant stable homotopy theory for compact Lie groups, which encompasses, and sheds a new light on, equivariant Atiyah duality and the Wirthmüller isomorphism.<br /> In Part I, we take a homotopy-theoretic approach, defining twisted ambidexterity in a general parametrized setup via a form of assembly map <em>f</em>_!(- &otimes; <em>D</em>_<em>f</em>) &rarr; <em>f</em>_*. When applied to equivariant homotopy theory for a compact Lie group <em>G</em>, we show that parametrized genuine <em>G</em>-spectra form the universal theory of stable <em>G</em>-equivariant objects which satisfy twisted ambidexterity for the orbits <em>G</em>/<em>H</em>. In simple terms, this says that genuine equivariant spectra differ from naive equivariant spectra only by the existence of Wirthmüller isomorphisms. <br /> In Part II, we take a differential-geometric approach, following ideas from motivic homotopy theory. We introduce for every separated differentiable stack X an &infin;-category SH(X) of genuine sheaves of spectra on X, which for a smooth manifold returns ordinary sheaves of spectra and for the classifying stack of a compact Lie group returns genuine equivariant spectra. We prove a form of relative Poincaré duality in this setting: for a proper representable submersion <em>f</em> of separated differentiable stacks, there is an equivalence <em>f</em>_<em>#</em> &cong; <em>f</em>_* (- &otimes; <em>S</em>^<em>Tf</em>) between its relative homology and a twist of its relative cohomology by the relative tangent sphere bundle. When specialized to quotient stacks of equivariant smooth manifolds, this recovers both equivariant Atiyah duality and the Wirthmüller isomorphism in stable equivariant homotopy theory.