For discrete groups G, we introduce equivariant Nielsen invariants. They are equivariant analogs of the Nielsen number and give lower bounds for the number of fixed point orbits in the G-homotopy class of an equivariant endomorphism f:X->X. Under mild hypotheses, these lower bounds are sharp.
We use the equivariant Nielsen invariants to show that a G-equivariant endomorphism f is G-homotopic to a fixed point free G-map if the generalized equivariant Lefschetz invariant of f is zero. Finally, we prove a converse of the equivariant Lefschetz fixed point theorem.
There is the problem, whether for a given virtually torsionfree discrete group $Γ$ there exists a cocompact proper topological $Γ$-manifold, which is equivariantly homotopy equivalent to the classifying space for proper actions. It is related to Nielsen's Realization and to the problem of Brown, whether there is a d-dimensional model for the classifying space for proper actions, if the underlying group has virtually cohomological dimension d. Assuming that the expected manifold model has a zero-dimensional singular set, we solve the problem in the Poincaré category and obtain new results about…
For $G$ a finite group, we show that functions on fields for the 2-dimensional supersymmetric sigma model with background $G$-symmetry determine cocycles for complex analytic $G$-equivariant elliptic cohomology. Similar structures in supersymmetric mechanics determine cocycles for equivariant K-theory with complex coefficients. The path integral for gauge theory with a finite group constructs wrong-way maps associated to group homomorphisms. When applied to an inclusion of groups, we obtain the induced character formula of Hopkins, Kuhn, and Ravenel. For the homomorphism $G\to *$ we obtain Vaf…
We construct a 2-category version of tom Dieck's equivariant fundamental groupoid for representable orbifolds and show that the discrete fundamental groupoid is Morita invariant; hence an orbifold invariant for representable orbifolds.
The first purpose of this paper is to examine the relationship between equivariant elliptic genera and orbifold elliptic genera. We apply the character theory of Hopkins et. al. to the Borel-equivariant genus associated to the sigma orientation of Ando-Hopkins-Strickland to define an orbifold genus for certain total quotient orbifolds and supersingular elliptic curves. We show that our orbifold genus is given by the same sort of formula as the orbifold ``two-variable'' genus of Dijkgraaf et al. In the case of a finite cyclic orbifold group, we use the characteristic series for the two-variable…
We define equivariant projective unitary stable bundles as the appropriate twists when defining K-theory as sections of bundles with fibers the space of Fredholm operators over a Hilbert space. We construct universal equivariant projective unitary stable bundles for the orbit types, and we use a specific model for these local universal spaces in order to glue them to obtain a universal equivariant projective unitary stable bundle for discrete and proper actions. We determine the homotopy type of the universal equivariant projective unitary stable bundle, and we show that the isomorphism classe…
We introduce the universal functorial equivariant Lefschetz invariant for endomorphisms of finite proper G-CW-complexes, where G is a discrete group. We use K_0 of the category of "phi-endomorphisms of finitely generated free RPi(G,X)-modules". We derive results about fixed points of equivariant endomorphisms of cocompact proper smooth G-manifolds.
We use a spectral sequence developed by Graeme Segal in order to understand the twisted G-equivariant K-theory for proper and discrete actions. We show that the second page of this spectral sequence is isomorphic to a version of Bredon cohomology with local coefficients in twisted representations. We furthermore explain some phenomena concerning the third differential of the spectral sequence, and we recover known results when the twisting comes from finite order elements in discrete torsion.
In this work the equivariant signature of a manifold with proper action of a discrete group is defined as an invariant of equivariant bordisms. It is shown that the computation of this signature can be reduced to its computation on fixed points sets equipped with their tubular neighborhoods. It is given a description of the equivariant vector bundles with action of a discrete group $G$ for the case when the action over the base is proper quasi-free, i.e. the stationary subgroup of any point is finite. The description is given in terms of some classifying space.
If K is a discrete group and Z is a K-spectrum, then the homotopy fixed point spectrum Z^{hK} is Map_*(EK_+, Z)^K, the fixed points of a familiar expression. Similarly, if G is a profinite group and X is a discrete G-spectrum, then X^{hG} is often given by (H_{G,X})^G, where H_{G,X} is a certain explicit construction given by a homotopy limit in the category of discrete G-spectra. Thus, in each of two common equivariant settings, the homotopy fixed point spectrum is equal to the fixed points of an explicit object in the ambient equivariant category. We enrich this pattern by proving in a preci…
After we have given a survey on the Burnside ring of a finite group, we discuss and analyze various extensions of this notion to infinite (discrete) groups. The first three are the finite-G-set-version, the inverse-limit-version and the covariant Burnside group. The most sophisticated one is the fourth definition as the equivariant zero-th cohomotopy of the classifying space for proper actions. In order to make sense of this definition we define equivariant cohomotopy groups of finite proper equivariant CW-complexes in terms of maps between the sphere bundles associated to equivariant vector b…
In this paper we develop Algebraic Morse Theory for the case where a group acts on a free chain complex. Algebraic Morse Theory is an adaption of Discrete Morse Theory to free chain complexes.
We introduce and compare two approaches to equivariant homotopy theory in a topological or ordinary Quillen model category. For the topological model category of spaces, we generalize Piacenza's result that the categories of topological presheaves indexed by the orbit category of a fixed topological group $G$ and the category of $G$-spaces can be endowed with Quillen equivalent model category structures. We prove an analogous result for any cofibrantly generated model category and discrete group $G$, under certain conditions on the fixed point functors of the subgroups of $G$. These conditions…
In this paper we show that the known models for $(\infty, 1)$-categories can all be extended to equivariant versions for any discrete group $G$. We show that in two of the models we can also consider actions of any simplicial group $G$.
The author has already proven that the space $Δ(Π_n)/G$ is homotopy equivalent to a wedge of spheres of dimension $n-3$ for all natural numbers $n\geq 3$ and all subgroups $G\subset S_1\times S_{n-1}$. We construct an $S_1\times S_{n-1}$-equivariant acyclic matching on $Δ(Π_n)$ together with a description of its critical simplices. This is also a more elementary approach to determining the number of spheres. We also develop new methods for Equivariant Discrete Morse Theory by adapting the Patchwork Theorem and poset maps with small fibers from Discrete Morse Theory.
We give a general framework of equivariant model category theory. Our groups G, called Hopf groups, are suitably defined group objects in any well-behaved symmetric monoidal category V. For any V, a discrete group G gives a Hopf group, denoted I[G]. When V is cartesian monoidal, the Hopf groups are just the group objects in V. When V is the category of modules over a commutative ring R, I[G] is the group ring R[G] and the general Hopf groups are the cocommutative Hopf algebras over R. We show how all of the usual constructs of equivariant homotopy theory, both categorical and model theoretic, …
The results of a previous paper on the equivariant homotopy theory of crossed complexes are generalised from the case of a discrete group to general topological groups. The principal new ingredient necessary for this is an analysis of homotopy coherence theory for crossed complexes, using detailed results on the appropriate Eilenberg-Zilber theory from Tonks' thesis, and of its relation to simplicial homotopy coherence. Again, our results give information not just on the homotopy classification of certain equivariant maps, but also on the weak equivariant homotopy type of the corresponding equ…
Suppose one is given a discrete group G, a cocompact proper G-manifold M, and a G-self-map f of M. Then we introduce the equivariant Lefschetz class of f, which is globally defined in terms of cellular chain complexes, and the local equivariant Lefschetz class of f, which is locally defined in terms of fixed point data. We prove the equivariant Lefschetz fixed point theorem, which says that these two classes agree. As a special case, we prove an equivariant Poincare-Hopf Theorem, computing the universal equivariant Euler characteristic in terms of the zeros of an equivariant vector field, and …
We compare three definitions of the equivariant cohomological dimension of a group with operators, coming from Takasu, Adamson and Bredon relative group cohomologies, giving examples of strict inequality in all cases where it can occur. We prove and make use of Stallings--Swan type results which characterise the groups of equivariant cohomological dimension one. Some of our examples are relevant to Farber's problem which asks for an algebraic characterisation of the topological complexity of discrete groups. In particular, the topological complexity of a group is not in general given by a rela…
We define a partial ordering on the set Q = Q(M) of pairs of topes of an oriented matroid M, and show the geometric realization |Q| of the order complex of Q has the same homotopy type as the Salvetti complex of M. For any element e of the ground set, the complex |Qe| associated to the rank-one oriented matroid on {e} has the homotopy type of the circle. There is a natural free simplicial action of Z4 on |Q|, with orbit space isomorphic to the order complex of the poset Q(M,e) associated to the pointed (or affine) oriented matroid (M,e). If M is the oriented matroid of an arrangement A of line…
We use a well known problem in discrete and computational geometry (partitions of measures by $k$-fans) as a motivation and as a point of departure to illustrate many aspects, both theoretical and computational, of the problem of calculating the obstructions for the existence of equivariant maps. A variety of techniques are introduced and discussed with the emphasis on concrete and explicit calculations. This eventually leads (Theorems 18 and 19) to an almost exhaustive analysis of when such maps do or do not exist in the particular case of interest.
We formalize an equivariant version of Bestvina-Brady discrete Morse theory, and apply it to Vietoris-Rips complexes in order to exhibit finite universal spaces for proper actions for all asymptotically CAT(0) groups.
We construct an equivariant version of discrete Morse theory for simplicial complexes endowed with group actions. The key ingredient is a 2-categorical criterion for making acyclic partial matchings on the quotient space compatible with an overlaid complex of groups. We use the discrete flow category of any such compatible matching to build the corresponding Morse complex of groups. Our main result establishes that the development of the Morse complex of groups recovers the original simplicial complex up to equivariant homotopy equivalence.
We study equivariant sheaves over profinite spaces, where the group is also taken to be profinite. We resolve a serious deficit in the existing theory by constructing a good notion of equivariant presheaves, with a suitable equivariant sheafification functor. Using equivariant sheafification, we develop the general theory of equivariant sheaves of modules over a ring, give explicit constructions of infinite products and introduce an equivariant analogue of skyscraper sheaves. These results underlie recent work by the authors which proves that there is an algebraic model for rational G-spectra …
Let G be a countable discrete group and let M be a smooth proper cocompact G-manifold without boundary. The Euler operator defines via Kasparov theory an element, called the equivariant Euler class, in the equivariant K-homology of M. The universal equivariant Euler characteristic of M, which lives in a group U^G(M), counts the equivariant cells of M, taking the component structure of the various fixed point sets into account. We construct a natural homomorphism from U^G(M) to the equivariant KO-homology of M. The main result of this paper says that this map sends the universal equivariant Eul…
In a previous paper, we have constructed, for an arbitrary Lie group G and any of the fields F=R or C, a good equivariant cohomology theory KF_G^*(-) on the category of proper $G$-CW-complex and have justified why it deserved the label ``equivariant K-theory". It was shown in particular how this theory was a logical extension of the construction of Lück and Oliver for discrete groups and coincided with Segal's classical K-theory when G is a compact group and only finite G-CW-complexes are considered. Here, we compare our new equivariant K-theory with that of N.C. Phillips: it is shown how a na…
In this note, we study equivariant versions of Stolz' $R$-groups, the positive scalar curvature structure groups $R^{\rm spin}_n(X)^G$, for proper actions of discrete groups $G$. We define the concept of a fundamental groupoid functor for a $G$-space, encapsulating all the fundamental group information of all the fixed point sets and their relations. We construct classifying spaces for fundamental groupoid functors. As a geometric result, we show that Stolz' equivariant $R$-group $R^{\rm spin}_n(X)^G$ depends only on the fundamental groupoid functor of the reference space $X$. The proof covers…
Let $G$ be a discrete group. We prove that the category of $G$-posets admits a model structure that is Quillen equivalent to the standard model structure on $G$-spaces. As is already true nonequivariantly, the three classes of maps defining the model structure are not well understood calculationally. To illustrate, we exhibit some examples of cofibrant and fibrant posets and an example of a non-cofibrant finite poset.
Let $G$ be an infinite discrete group and let $\underline{E}G$ be a classifying space for proper actions of $G$. Every $G$-equivariant vector bundle over $\underline{E}G$ gives rise to a compatible collection of representations of the finite subgroups of $G$. We give the first examples of groups $G$ with a cocompact classifying space for proper actions $\underline{E}G$ admitting a compatible collection of representations of the finite subgroups of $G$ that does not come from a $G$-equivariant (virtual) vector bundle over $\underline{E}G$. This implies that the Atiyah-Hirzeburch spectral sequen…
