Finite Virtual Cohomological Dimension Research Articles

Milnor excision for motivic spectra

Abstract We prove that the ∞ {\infty} -category of motivic spectra satisfies Milnor excision: if A → B {A\to B} is a morphism of commutative rings sending an ideal I ⊂ A {I\subset A} isomorphically onto an ideal of B, then a motivic spectrum over A is equivalent to a pair of motivic spectra over B and A / I {A/I} that are identified over B / I ⁢ B {B/IB} . Consequently, any cohomology theory represented by a motivic spectrum satisfies Milnor excision. We also prove Milnor excision for Ayoub’s étale motives over schemes of finite virtual cohomological dimension.

The topological nilpotence degree of a Noetherian unstable algebra

We investigate the topological nilpotence degree, in the sense of Henn–Lannes–Schwartz, of a connected Noetherian unstable algebra R. When R is the mod p cohomology ring of a compact Lie group, Kuhn showed how this invariant is controlled by centralizers of elementary abelian p-subgroups. By replacing centralizers of elementary abelian p-subgroups with components of Lannes’ T-functor, and utilizing the techniques of unstable algebras over the Steenrod algebra, we are able to generalize Kuhn’s result to a large class of connected Noetherian unstable algebras. We show how this generalizes Kuhn’s result to more general classes of groups, such as groups of finite virtual cohomological dimension, profinite groups, and Kac–Moody groups. In fact, our results apply much more generally, for example, we establish results for p-local compact groups in the sense of Broto–Levi–Oliver, for connected H-spaces with Noetherian mod p cohomology, and for the Borel equivariant cohomology of a compact Lie group acting on a manifold. Along the way we establish several results of independent interest. For example, we formulate and prove a version of Carlson’s depth conjecture in the case of a Noetherian unstable algebra of minimal depth.

Depth and detection for Noetherian unstable algebras

For a connected Noetherian unstable algebra R R over the mod p p Steenrod algebra, we prove versions of theorems of Duflot and Carlson on the depth of R R , originally proved when R R is the mod p p cohomology ring of a finite group. This recovers the aforementioned results, and also proves versions of them when R R is the mod p p cohomology ring of a compact Lie group, a profinite group with Noetherian cohomology, a Kac–Moody group, a discrete group of finite virtual cohomological dimension, as well as for certain other discrete groups. More generally, our results apply to certain finitely generated unstable R R -modules. Moreover, we explain the results in the case of the p p -local compact groups of Broto, Levi, and Oliver, as well as in the modular invariant theory of finite groups.

The Motivic Hopf Map Solves the Homotopy Limit Problem for $K$-Theory

We solve affirmatively the homotopy limit problem for (effective) K -theory over fields of finite virtual cohomological dimension. Our solution uses the motivic slice filtration and the first motivic Hopf map.

Inclusion of configuration spaces in Cartesian products, and the virtual cohomological dimension of the braid groups of 𝕊2 and ℝP2

Let M be a surface, perhaps with boundary, and either compact, or with a finite number of points removed from the interior of the surface. We consider the inclusion i: F\_n(M) --\textgreater{} M^n of the nth configuration space F\_n(M) of M into the n-fold Cartesian product of M, as well as the induced homomorphism i\_\#: P\_n(M) --\textgreater{} (\pi\_1(M))^n, where P\_n(M) is the n-string pure braid group of M. Both i and i\_\# were studied initially by J.Birman who conjectured that Ker(i\_\#) is equal to the normal closure of the Artin pure braid group P\_n in P\_n(M). The conjecture was later proved by C.Goldberg for compact surfaces without boundary different from the 2-sphere S^2 and the projective plane RP^2. In this paper, we prove the conjecture for S^2 and RP^2. In the case of RP^2, we prove that Ker(i\_\#) is equal to the commutator subgroup of P\_n(RP^2), we show that it may be decomposed in a manner similar to that of P\_n(S^2) as a direct sum of a torsion-free subgroup L\_n and the finite cyclic group generated by the full twist braid, and we prove that L\_n may be written as an iterated semi-direct product of free groups. Finally, we show that the groups B\_n(S^2) and P\_n(S^2) (resp. B\_n(RP^2) and P\_n(RP^2)) have finite virtual cohomological dimension equal to n-3 (resp. n-2), where B\_n(M) denotes the full n-string braid group of M. This allows us to determine the virtual cohomological dimension of the mapping class groups of the mapping class groups of S^2 and RP^2 with marked points, which in the case of S^2, reproves a result due to J.Harer.

The Yagita Invariant of Some Symplectic Groups

If a group G of finite virtual cohomological dimension has p-periodic Farrell cohomology, then the Yagita invariant of this group equals its p-period. The group of symplectic 2(p + 1) × 2(p + 1) matrices over \({\mathbb{Z}}\) has elementary abelian p-subgroups of rank at least 2 and hence \({{\rm Sp}(2(p+1), \mathbb{Z})}\) and \({{\rm Sp}(2(p+1), \mathbb{Q})}\) do not have p-periodic Farrell cohomology. We compute the Yagita invariants of these groups for any odd prime p.

Delta-discreteG–spectra and iterated homotopy fixed points

Let G be a profinite group with finite virtual cohomological dimension and let X be a discrete G-spectrum. If H and K are closed subgroups of G, with H normal in K, then, in general, the K/H-spectrum X^{hH} is not known to be a continuous K/H-spectrum, so that it is not known (in general) how to define the iterated homotopy fixed point spectrum (X^{hH})^{hK/H}. To address this situation, we define homotopy fixed points for delta-discrete G-spectra and show that the setting of delta-discrete G-spectra gives a good framework within which to work. In particular, we show that by using delta-discrete K/H-spectra, there is always an iterated homotopy fixed point spectrum, denoted (X^{hH})^{h_\delta K/H}, and it is just X^{hK}. Additionally, we show that for any delta-discrete G-spectrum Y, (Y^{h_\delta H})^{h_\delta K/H} \simeq Y^{h_\delta K}. Furthermore, if G is an arbitrary profinite group, there is a delta-discrete G-spectrum {X_\delta} that is equivalent to X and, though X^{hH} is not even known in general to have a K/H-action, there is always an equivalence ((X_\delta)^{h_\delta H})^{h_\delta K/H} \simeq (X_\delta)^{h_\delta K}. Therefore, delta-discrete L-spectra, by letting L equal H, K, and K/H, give a way of resolving undesired deficiencies in our understanding of homotopy fixed points for discrete G-spectra.

Function spectra and continuous G -spectra

Let G be a profinite group, {Xα}α be a cofiltered diagram of discrete G-spectra, and Z be a spectrum with trivial G-action. We show how to define the homotopy fixed point spectrum F(Z, holimα Xα)hG and that when G has finite virtual cohomological dimension (vcd), it is equivalent to F(Z, holimα (Xα)hG). With these tools, we show that the K(n)-local Spanier–Whitehead dual is always a homotopy fixed point spectrum, a well-known Adams-type spectral sequence is actually a descent spectral sequence, and, for a sufficiently nice k-local profinite G-Galois extension E, with K ◃ G and closed, the equivalence ( E h k K ) h k G / K ≃ E h k G (due to Behrens and the author), where ( − ) h k ( − ) denotes k-local homotopy fixed points, can be upgraded to an equivalence that just uses ordinary (non-local) homotopy fixed points, when G/K has finite vcd.

When is group cohomology finitary?

If G is a group, then we say that the functor H n ( G , − ) is finitary if it commutes with all filtered colimit systems of coefficient modules. We investigate groups with cohomology almost everywhere finitary; that is, groups with nth cohomology functors finitary for all sufficiently large n. We establish sufficient conditions for a group G possessing a finite-dimensional model for E ̲ G to have cohomology almost everywhere finitary. We also prove a stronger result for the subclass of groups of finite virtual cohomological dimension, and use this to answer a question of Leary and Nucinkis. Finally, we show that if G is a locally (polycyclic-by-finite) group, then G has cohomology almost everywhere finitary if and only if G has finite virtual cohomological dimension and the normalizer of every non-trivial finite subgroup of G is finitely generated.

Groups possessing extensive hierarchical decompositions

Kropholler's class of groups is the smallest class of groups which contains all finite groups and is closed under the following operator: whenever $G$ admits a finite-dimensional contractible $G$-CW-complex in which all stabilizer groups are in the class, then $G$ is itself in the class. Kropholler's class admits a hierarchical structure, i.e., a natural filtration indexed by the ordinals. For example, stage 0 of the hierarchy is the class of all finite groups, and stage 1 contains all groups of finite virtual cohomological dimension. We show that for each countable ordinal $\alpha$, there is a countable group that is in Kropholler's class which does not appear until the $\alpha+1$st stage of the hierarchy. Previously this was known only for $\alpha= 0$, 1 and 2. The groups that we construct contain torsion. We also review the construction of a torsion-free group that lies in the third stage of the hierarchy.

ON GROUPS OF TYPE $\mathcal{L}^{-1}$

We show that every finite group G has a set of cohomological elements satisfying ceratin algebraic property [Formula: see text] which can be regarded as a generalized notion of an algebraic counterpart to the topological phenomenon of free actions on finite dimensional homotopy spheres. We extend this result to a certain class of groups which contains groups of finite virtual cohomological dimension.

Finiteness properties of automorphism groups of right-angled Artin groups

We study the algebraic structure of the outer automorphism group of a general right-angled Artin group. We show that this group is virtually torsion-free and has finite virtual cohomological dimension. This generalizes results proved earlier by the authors and Crisp for 2-dimensional right-angled Artin groups.

On the Zeroeth Complete Cohomology of Certain Polycyclic Groups

The zeroeth complete cohomology of a group G with coefficients in , is known to carry much information about the group. Its computation is rather complicated. Kropholler (1993b) and Benson and Kropholler (1995) stated that even when G is a polycyclic by finite group, no general formula is known for . Here we construct a projective resolution for a class of polycyclic groups and obtain information about , which indicates that no general formula for can be expected. Moreover, some groups in our class are counterexamples to a conjecture of Brown, which states that the least common multiple of the orders of the finite subgroups of a group of finite virtual cohomological dimension annihilates the complete cohomology of G. The first counterexample to this conjecture was given by Adem (1989) and more were obtained later by Adem and Carlson (1990) and Leary (1996). We obtain counterexamples that are not in those classes.

Explicit fibrant replacement for discrete $G$-spectra

If C is the model category of simplicial presheaves on a site with enough points, with fibrations equal to the global fibrations, then it is well-known that the fibrant objects are, in general, mysterious. Thus, it is not surprising that, when G is a profinite group, the fibrant objects in the model category of discrete G-spectra are also dicult to get a handle on. However, with simplicial presheaves, it is possible to construct an explicit fibrant model for an object in C, under certain finiteness conditions. Similarly, in this paper, we show that if G has finite virtual cohomological dimension and X is a discrete Gspectrum, then there is an explicit fibrant model for X. Also, we give several applications of this concrete model related to closed subgroups of G.

On a relation between certain cohomological invariants

Let G be a group, spli Z G the supremum of the projective lengths of the injective Z G -modules and silp Z G the supremum of the injective lengths of the projective Z G -modules. The invariants spli Z G and silp Z G were studied in [T.V. Gedrich, K.W. Gruenberg, Complete cohomological functors on groups, Topology Appl. 25 (1987) 203–223] in connection with the existence of complete cohomological functors. If spli Z G is finite then silp Z G = spli Z G = findim Z G [T.V. Gedrich, K.W. Gruenberg, Complete cohomological functors on groups, Topology Appl. 25 (1987) 203–223] and cd ¯ Z G ≤ spli Z G ≤ cd ¯ Z G + 1 , where cd ¯ Z G is the generalized cohomological dimension of G [B.M. Ikenaga, Homological dimension and Farrell cohomology, J. Algebra 87 (1984) 422–457]. Note that cd ¯ Z G = vcd G if G is of finite virtual cohomological dimension. It has been conjectured in [O. Talelli, On groups of type Φ , Arch. Math. 89 (1) (2007) 24–32] that if spli Z G is finite then G admits a finite dimensional model for E ¯ G , the classifying space for proper actions. We conjecture that spli Z G = cd ¯ Z G + 1 for any group G and we prove the conjecture for duality groups, fundamental groups of graphs of finite groups and fundamental groups of certain finite graphs of groups of type FP ∞ .

Automorphisms of 2–dimensional right-angled Artin groups

We study the outer automorphism group of a right-angled Artin group A_G in the case where the defining graph G is connected and triangle-free. We give an algebraic description of Out(A_G) in terms of maximal join subgraphs in G and prove that the Tits' alternative holds for Out(A_G). We construct an analogue of outer space for Out(A_G) and prove that it is finite dimensional, contractible, and has a proper action of Out(A_G). We show that Out(A_G) has finite virtual cohomological dimension, give upper and lower bounds on this dimension and construct a spine for outer space realizing the most general upper bound.

Permutation complexes for profinite groups

An important tool in the analysis of discrete groups of finite virtual cohomological dimension is the existence of a finite dimensional contractible CW-complex on which the group acts with finite stabilizers. We develop a purely algebraic analogue for profinite groups. This enables us to reveal the connection between finiteness conditions on the cohomology of the group and those on the normalizers of the finite p-subgroups.

On cohomology isomorphisms of groups

We show how Mislin's theorem on group homomorphisms that induce an isomorphism in cohomology can be proved based on ideas of Alperin and the use of Lannes's T-functor. This enables us to extend the class of groups to include groups of finite virtual cohomological dimension and profinite groups.

The outer space of a free product

We associate a contractible ‘outer space’ to any free product of groups G = G1 * … * Gq. It is identical to Culler–Vogtmann space when G is free, and McCullough–Miller space when no Gi is ℤ. Our proof of contractibility (given when G is not free) is based on Skora's idea of deforming morphisms between trees. Using the action of Out(G) on this space, we show that Out(G) has finite virtual cohomological dimension, or is VFL (it has a finite index subgroup with a finite classifying space), if the groups Gi and Out(Gi) have similar properties. We deduce that Out(G) is VFL if G is a torsion-free hyperbolic group, or a limit group (finitely generated fully residually free group).

Some Novikov rings that are von Neumann finite and knot-like groups

We show that for a finitely generated group G and for every discrete character \chi\colon G \rightarrow \mathbb{Z} any matrix ring over the Novikov ring \widehat{\mathbb{Z}G}_{\chi} is von Neumann finite. As a corollary we obtain that if G is a non-trivial discrete group with a finite K(G,1) CW-complex Y of dimension n and Euler characteristics zero and N is a normal subgroup of G of type FP_{n-1} containing the commutator subgroup G' and such that G/N is cyclic-by-finite, then N is of homological type FP_n and G/N has finite virtual cohomological dimension vcd(G/N) = cd(G) - cd(N). This completes the proof of the Rapaport Strasser conjecture that for a knot-like group G with a finitely generated commutator subgroup G' the commutator subgroup G' is always free and generalises an earlier work by the author where the case when G' is residually finite was proved. Another corollary is that a finitely presentable group G with def(G) > 0 and such that G' is finitely generated and perfect can be only \mathbb{Z} or \mathbb{Z}^2 , a result conjectured by A. J. Berrick and J. Hillman in [1].