geometric representation theory
representation, 2-representation, ∞-representation
Grothendieck group, lambda-ring, symmetric function, formal group
principal bundle, torsor, vector bundle, Atiyah Lie algebroid
Eilenberg-Moore category, algebra over an operad, actegory, crossed module
Be?linson-Bernstein localization?
In the context of equivariant stable homotopy theory and $G$ one distinguishes, for a G-spectrum $E$, between the plain fixed point spectrum $F^G(E)$ and its geometric fixed point spectrum $\Phi^G(E)$.
Here the terminology “geometric” is in the sense of point-set topology, as opposed to homotopy theory: If $X$ is a (pointed) topological space equipped with a continuous function $G$-action (a topological G-space), so that one may consider its $G$-equivariant suspension spectrum $\Sigma^\infty_G X \in G Spectra$, then the geometric fixed point spectrum $\Phi^G(\Sigma^\infty_G X)$ of the latter is equivalent to the plain suspension spectrum of the plain fixed point-space $X^G$:
see the characterization in Prop. , below.
In general this is different from (not equivalent to) both of the following other notions of fixed point spectra:
the plain (really: homotopy theoretic) fixed point spectrum $F^G(\Sigma^\infty_G X)$, which is instead the derived functor of forming topological fixed points $X \mapsto X^G$, hence which applies this construction only after fibrant resolution; the difference between the two is described by the tom Dieck splitting theorem, see Prop. below.
the categorical fixed point spectrum?…
A concrete definition of geometric fixed points of an equivariant spectrum is in (Schwede 15, 7.3). A conceptual characterization in terms of localization of spectra is in (Mathew-Naumann-Noel 15, def. 6.12).
(as a wedge summand in the tom Dieck splitting)
For $X$ a topological G-space and $\Sigma^\infty_G X$ its equivariant suspension spectrum, there is a canonical comparison morphism (…)
which exhibits its geometric fixed point spectrum as precisely the first summand in the tom Dieck splitting of the plain fixed point spectrum
In fact:
The construction of geometric fixed point spectra is essentially uniquely characterized by the property
and the property of being left derived strong monoidal and preserving homotopy colimits.
(Schwede 15, remark 7.15, Blumberg 17, around Def. 2.5.16)
More generally:
(partial geometric fixed point spectra)
There is a “partial” geometric fixed point functor, which for a given subgroup $H \subset G$ sends
(for $W_G/H$ the Weyl group, which is the quotient group $G/H$ in the case that $H$ is a normal subgroup) and satisfies for a $G$-equivariant suspension spectrum $\Sigma^\infty_G X$ the relation
hence, if $H = N \subset G$ already happens to be a normal subgroup:
(Lewis-May-Steinberger 86, II.9, Def. 9.7, Cor. 9.9, Lewis 00, Scholium 10.2)
$\,$
We collect some facts from Lewis-May-Steinberger 86, section II.9.
Throughout, consider a finite group $G$ and a normal subgroup $N \subset G$.
We write
for the set of subgroups of $G$ that do not contain $N$, and
for the subset of subgroups of $G$ that do contain $N$.
(LMS86, p. 107 & bottom of p. 109)
whose fixed point spaces for subgroups $H \subset G$ are
We say that a morphism $f \colon X \to Y$ of G-spectra is an $\mathcal{F}[N]^'$-equivalence if its smash product with $\tilde E \mathcal{F}[N]$ (Def. )
is an equivalence of G-spectra.
The localization of $G Spectra$ at the $\mathcal{F}[N]'$-equivalences (Def. ) is a smashing localization, given by smashing with the equivariant suspension spectrum of $\tilde E \mathcal{F}[N]$ (Def. )
In particular, we have
(Lewis-May-Steinberger 86, Prop. II 9.1, 9.2 & top of p. 109)
Hence
is $\mathcal{F}'[N]$-localization on hom-objects.
For $X$ and $Y$ G-CW-complexes, the following are bijections of hom-sets:
(LMS 86, prop. II 9.3 with remark below the proof)
On hom-sets of G-spaces $Ho_{G Spaces}(X,Y)$, postcomposing with the smashing $(S^0 \to \tilde E \mathcal{F}[N]) \wedge Y$ is isomorphic to restricting along $X^N \hookrightarrow X$: The following is a commuting square (by nature of the hom-functor) and the right and bottom morphisms are bijections by Lemma :
(geometric fixed point spectra in terms of homotopy fix point spectra)
The partial geometric fixed point functor (Prop. )
is given on equivariant suspension spectra $\Sigma^\infty_G X$ equivalently by first smashing with $\tilde E \mathcal{F}[N]$ (Def. ) followed by forming the partial plain fixed point spectrum:
(Lewis-May-Steinberger 86, Cor. 9.9)
We will also need this here:
For $X$ a G-CW-complex $E$ a G- CW-spectrum and $N \subset G$ a normal subgroup, the partial $N$-fixed point spectrum functor on spectra and the plain fixed point functor on spaces are compatible with smash product:
(Lewis-May-Steinberger 86, prop. II 9.12)
We discuss an explicit formula (Prop. below, due to Lewis-May-Steinberger 86) that expresses equivariant cohomology groups with coefficients in partial geometric fixed point spectra (Prop. ) as the equivariant cohomology groups with coefficients in the original spectrum, but with certain “Euler classes inverted”.
As an application, we show (Example below) that the equivariant stable cohomotopy of the point in certain non-trivial RO(G)-degrees $V$ surjects onto the corresponding partially equivariant stable cohomotopy in degree 0 (the latter being well-understood: given by the Burnside ring, by this Prop).
$\,$
A key role in this discussion is played by those RO(G)-degrees which trivialize when passing to partial fixed points:
(RO(G)-degrees without non-trivial $H$-fixed points)
For $H \subset G$ a subgroup, say that a $G$-representation $V$ has no non-trivial $H$-fixed points if the fixed point space of $V$ with respect to the $H$-action is the origin (which is necessarily fixed), hence is the zero-representation:
We also use the following notation, following Lewis-May-Steinberger 86:
(base change along normal subgroup-inclusions of equivariance-groups)
Given a normal subgroup-inclusion
with induced projection $\epsilon$ to the quotient group $G/N$ this induces various base change adjunctions (on homotopy categories, say), such as on topological G-spaces, to be denoted
and on $G$-representations, to be denoted
and on G-spectra, to be denoted
where the right adjoint $(-)^N$ is the partial fixed point spectrum-functor (in contrast to the geometric fixed point functor).
(e.g. Lewis-May-Steinberger 86, above theorem 9.5)
(partial geometric fixed point cohomology via inversion of Euler classes)
Let $E \;\in\; G Spectra$ be a G-spectrum with partial geometric fixed point spectrum $\Phi^N E \;\in\; G/N Spectra$ (Prop. ) and let $X \;\in\; G/N Spectra^{fin} \overset{\epsilon^\sharp}{\longrightarrow} G Spectra$ be finite $G$-CW-spectrum.
Then the $G/N$-equivariant cohomology groups in RO(G/N)-degree $\alpha$ of $X$ with coefficients in the partial geometric fixed point spectrum $\Phi^N E$ are equivalently the colimit over the $G$-equivariant cohomology groups of $\epsilon^\sharp X$ (2) with coefficients in $E$, but in RO(G)-degree $\epsilon^\ast \alpha$ (3) plus a shift by all those representations $V$ that have no nontrivial $N$-fixed points (Def. ):
where the colimit is over the diagram that has precisely one morphism for every inclusion $V_1 \subset V_2$ of $G$-representations without non-trivial $N$-fixed points (Def. )
given by smash product with the Euler class
of $V \coloneqq V_2 - V_1$.
(Lewis-May-Steinberger 86, chapter II, prop. 9.13)
(comparison map to partial geometric fixed point cohomology)
Prop. provides a canonical comparison morphism, to be denoted
from the $G$-equivariant cohomology groups with coefficients in $E$ to those with coefficients in the partial geometric fixed point spectrum: Namely the component of the colimiting cocone(5) at stage $V = 0$:
This component is equal to the following composite of isomorphisms with $\mathcal{F}[N]'$-localization $L_{\mathcal{F}[N]'}$ (Def. ):
This follows from the proof of (Lewis-May-Steinberger 86, chapter II, prop. 9.13). We make this explicit: The proof there says that the comparison map is given by the smashing with $S^0 \to \tilde E \mathcal{F}$, up to re-identifications:
The first equality in (7) is the definition of cohomology classes;
the second step is the unitor isomorphism for the tensor unit being the sphere spectrum;
the third step is smashing with $S^0 \to \tilde E \mathcal{F}[N]$, which is $\mathcal{F}'$-localization by Prop. ;
the fourth step is the hom-isomorphism for the adjunction $( \epsilon^\sharp \dashv (-)^N )$ from (4);
the sixth step is the evident identification $(S^{\epsilon^\ast \alpha})^N = S^\alpha$ in the first smash factor, and is Lemma in the second factor.
the seventh step is again the definition of cohomology.
$\,$
(equivariant stable cohomotopy of the point in non-trivial RO(G)-degree)
Let $G$ be a finite group. Then the canonical comparison morphism (6) from Def. exhibits the $G$-equivariant stable cohomotopy group of the point in any RO(G)-degree $V$ that has trivial $N$-fixed points ($V^N = 0$, Def. ) as a group extension of the $G/N$-equivariant stable cohomotopy of the point in RO(G/N)-degree zero, hence of the group underlying the Burnside ring $A(G/N)$ (this Prop.):
First observe that, in the given situation, the comparison morphism $p_{\mathbb{S}}^N(\ast)$ (6) is indeed of the form shown, up to isomorphism: We are in the situation of Prop. for
$X \coloneqq \Sigma^\infty_{G/N}(\ast_+) = \Sigma^\infty_{G/N} S^0$, which is clearly a finite $G/N$-CW-spectrum;
$E \coloneqq \Sigma^V_G\mathbb{S}_G \coloneqq \Sigma^\infty_G S^V$ the $V$-shifted $G$-equivariant sphere spectrum, being the G-spectrum representing $G$-equivariant stable cohomotopy, by definition;
$\Phi^N E \simeq \Sigma^\infty_{G/N} (S^V)^N \simeq \Sigma^\infty_{G/N} S^0 \simeq \mathbb{S}_{G/N}$ the unshifted $G/N$-equivariant sphere spectrum, by (1) and by assumption on $V$.
Hence with all identifications made explicit, the morphism (8) in question is the composite
of $p_{ \Sigma^V_G \mathbb{S}_G }^N(\ast)$ with a sequence of isomorphisms, and hence our task is to prove that $p_{ \Sigma^V_G \mathbb{S}_G }^N(\ast)$ is a surjection.
We first prove this for the case that $V = 0$. In this case the identification with the Burnside ring (via this Prop.) applies also to the domain cohomology group:
By Prop. the comparison morphism acts on this by smashing the codomain of the hom-sets with $(S^0 \to \tilde E \mathcal{F}[N])$. But by Corollary this is equivalent to restricting to $N$-fixed point spaces so that (8) becomes simply the projection of Burnside rings
sending any G-set $K$ to its subset $K^N$ of $N$-fixed points regarded with its residual $G/N$-action.
This is clearly surjective. (The irreducible elements on the right are the isomorphism classes of the transitive $G/N$-actions $(G/N)/H$ for $H \subset G/H$, which are canonically also G-sets, hence have a pre-image on the left.)
In order to deduce the general statement from this special case, we now make use of the fact that Prop. says that the comparison map for $V = 0$ is one coprojection map of a colimiting cocone-diagram, which for each $G$-representation $V$ without non-trivial $N$-fixed points (Def. ) contains a cocone component of the following form:
Since we know, as just argued, that the map $(-)^N$ on the left is a surjection, the commutativity of this diagram implies that also the component projection $p_V$ is surjective. (Every element $c \in A(G/N)$ has a lift to $\widehat c \in A(G)$, but then the commutativity of the triangle means that $\widehat c \wedge \chi_V$ is a pre-image of $c$ under $p_V$.)
This we may use to deduce the statement for the general case, where the codomain of (9) is in degree $V$:
By the assumption that the RO(G)-degree $V$ has no non-trivial $N$-fixed points, Prop. says that the colimiting cocone in which the map $p_{ \Sigma^V_G \mathbb{S}_G }^N(\ast)$ appears, by Def. , looks just like the one above, except that it “starts” not in degree 0, but in degree $V$:
In particular the cocone in (10) restricts to a cocone over this sub-diagram in (11), so that the universal property of the cocone in (11) implies an endomorphism $\phi$ of the abelian group underlying the Burnside ring $(\Sigma^\infty_G S^0)^0_{G/N}(\ast) = A(G/n)$ such that
Since $p_V$ is surjective, it is now sufficient to prove that this $\phi$ is in fact an isomorphism.
To see this, observe that, since $G$ is a finite group by assumption, the abelian group underlying the Burnside ring $A(G/N)$ is a finitely generated free abelian group (spanned by the cosets $(G/N)/H$ as $H$ ranges over the finite set of conjugacy classes of subgroups of $G/N$ ). By the structure theory of free abelian groups, this means that $\phi$ may be represented by a matrix in Smith normal form. Specifically, since $\phi$ is an endomorphism, it is represented by a square matrix in Smith normal form. Since $\phi \circ p_{ \Sigma^V_G \mathbb{S}_G }^N(\ast)$ is surjective, by (12) and the surjectivity of $p_V$ established before, this implies that $\phi$ is represented by a diagonal matrix all whose diagonal entries are non-vanishing and invertible, hence that $\phi$ is in fact an isomorphism.
With this, (12) says that with $p_V$ also $p_{ \Sigma^V_G \mathbb{S}_G }^N(\ast)$ is surjective.
categorical fixed point spectrum?
L. Gaunce Lewis, Peter May, and Mark Steinberger (with contributions by J.E. McClure), Equivariant stable homotopy theory, Springer Lecture Notes in Mathematics Vol.1213. 1986 (pdf)
L. Gaunce Lewis, Jr., section 10 of Splitting theorems for certain equivariant spectra, Memoirs of the AMS, number 686, March 2000, Volume 144 (pdf)
Stefan Schwede, section 7.3 of Lectures on Equivariant Stable Homotopy Theory, 2015 (pdf)
Stefan Schwede, section 3.3. of Global homotopy theory (arXiv:1802.09382)
Andrew Blumberg, Def. 2.5.16 in Equivariant homotopy theory, 2017 (pdf, GitHub)
Akhil Mathew, Niko Naumann, Justin Noel, Nilpotence and descent in equivariant stable homotopy theory (arXiv:1507.06869)
Tom Bachmann, Marc Hoyois, remark 9.9 in Norms in motivic homotopy theory (arxiv:1711.03061)
Relation to spectral Mackey functors: