We give a new construction of the equivariant -theory of group actions of Barwick et al., by producing an infinite loop -space for each Waldhausen category with -action, for a finite group . On the category of retractive spaces over a -space , this produces an equivariant lift of Waldhausen’s functor , and we show that the -fixed points are the bivariant -theory of the fibration . We then use the framework of spectral Mackey functors to produce a second equivariant refinement whose fixed points have tom Dieck type splittings. We expect this second definition to be suitable for an equivariant generalization of the parametrized -cobordism theorem.
Waldhausen’s celebrated construction, and the “parametrized -cobordism” theorem relating it to the space of -cobordisms on , provides a critical link in the chain of homotopy-theoretic constructions relating the behavior of compact manifolds to that of their underlying homotopy types [waldhausen1978alg] [waldhausennew]. While the -theory assembly map provides the primary invariant that distinguishes the closed manifolds in a given homotopy type, provides the secondary information that accesses the diffeomorphism and homeomorphism groups in a stable range [ww]. And in the case of compact manifolds up to stabilization, accounts for the entire difference between the manifold and its underlying homotopy type with tangent information [dww]. As a consequence, calculations of have immediate consequences for the automorphism groups of high-dimensional closed manifolds, and of compact manifolds up to stabilization.
When the manifolds in question have an action by a group , there is a similar line of attack for understanding the equivariant homeomorphisms and diffeomorphisms. One expects to replace with an appropriate space of -isovariant -cobordisms on , stabilized with respect to representations of . The connected components of such a space would be expected to coincide with the equivariant Whitehead group of [luck], which splits as
where denotes conjugacy classes of subgroups. This splitting is reminiscent of the tom Dieck splitting for genuine -suspension spectra
and suggests that the variant of -theory most directly applicable to manifolds will in fact be a genuine -spectrum, whose fixed points have a similar splitting.
In this paper we begin to realize this conjectural framework. We define an equivariant generalization of Waldhausen’s -theory functor, when is a space with an action by a finite group , whose fixed points have the desired tom Dieck style splitting.
Theorem 1.1 (LABEL:agx_exists2@).
For a finite group, there exists a functor from -spaces to genuine -spectra with fixed points
and a similar formula for the fixed points of each subgroup .
To be more specific, the fixed points are the -theory of the category of finite retractive -cell complexes over , with equivariant weak homotopy equivalences between them. The splitting of this -theory is a known consequence of the additivity theorem, and an explicit proof appears in [wojciech].
In a subsequent paper, we plan to explain how fits into a genuinely -equivariant generalization of Waldhausen’s parametrized -cobordism theorem. The argument we have in mind draws significantly from an analysis of the fixed points of our carried out by Badzioch and Dorabiała [wojciech], and a forthcoming result of Goodwillie and Igusa that defines and gives a splitting that recovers (1). We emphasize that lifting these theorems to genuine -spectra permits the tools of equivariant stable homotopy theory to be applied to the calculation of , in addition to the linearization and trace techniques that have been used so heavily in the nonequivariant case.
Most of the work in this paper is concerned with constructing equivariant spectra out of category-theoretic data. One approach is to generalize classical delooping constructions such as the operadic machine of May [MayGeo] or the -space machine of Segal [segal] to allow for deloopings by representations of . Using the equivariant generalization of the operadic infinite loop space machine from [GM3], we show how this approach generalizes to deloop Waldhausen -categories.
The theory of Waldhausen categories with -action is subtle. Even when the -action is through exact functors, the fixed points of such a category do not necessarily have Waldhausen structure (LABEL:waldfixedpts@). Define be the category with objects the elements of and precisely one morphism between any two objects, whose classifying space is . Let be the category of all functors and all natural transformations with acting by conjugation; we define the homotopy fixed points of a -category as the fixed point category , and we explain in §LABEL:waldhausen_gcat how this category does have a Waldhausen structure.
The “equivariant -theory of group actions” of Barwick, Glasman, and Shah produces a genuine -spectrum (using the framework of [Gmonster]) whose -fixed points are [Gmonster2, §8]. We complement this with a result that shows the -space may be directly, equivariantly delooped.
Theorem 1.2 (LABEL:inf_loop@ and LABEL:fixed_points_agree@).
If is a Waldhausen -category then the -theory space defined as , where is Waldhausen’s construction from [waldhausen], is an equivariant infinite loop space. The -fixed points of the resulting --spectrum are equivalent to the -theory of the Waldhausen category for every subgroup .
The downside of this approach is that one does not have much freedom to modify the weak equivalences in the fixed point categories. Note that if is a -space, then the category of homotopy finite retractive spaces over has a -action. For a retractive space , is defined by precomposing the inclusion map by and postcomposing the retraction map by . We can apply 1.2 to this category, and the resulting theory has as its -fixed points the -theory of -equivariant spaces over , as we expect, but the weak equivalences are the -maps which are nonequivariant homotopy equivalences. Thus, Theorem 1.2 does not suffice to prove Theorem 1.1.
Although does not match our expected input for the -cobordism theorem, it does have a surprising connection to the bivariant -theory of Williams [bruce]:
Theorem 1.3 (LABEL:prop:coarse_equals_bivariant@ and LABEL:thm:homotopy_fixed_equals_coassembly@).
There is a natural equivalence of spectra
Under this equivalence, the coassembly map for bivariant -theory agrees up to homotopy with the map from fixed points to homotopy fixed points: