Cohomological invariants and  the classifying space  for proper actions

Gandini, Giovanni

doi:10.4171/ggd/169

Cited by 14 publications

(12 citation statements)

References 23 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The Gorenstein cohomological dimension of

G

over

double-struckZ

enjoys several useful properties and is proposed in to serve as an algebraic invariant whose finiteness characterizes the groups which admit a finite‐dimensional model for the classifying space for proper actions. It is related to many other numerical invariants that are studied in cohomological group theory: As shown in , for any group

G

we have a chain of inequalities

{Gcd}_{double-struckZ} G ⩽ F {cd}_{double-struckZ} G ⩽ {cd}_{frakturF} G ⩽ {gd}_{frakturF} G ⩽ max false{3, {cd}_{F} G false} .

Here,

F {cd}_{double-struckZ} G

is the so‐called

frakturF

‐cohomological dimension of

G

over

double-struckZ

, which is defined in terms of relative homological algebra, relative to the class

frakturF

of finite subgroups of

G

. The Bredon cohomological dimension

{cd}_{frakturF} G

G

is defined in terms of the category of all contravariant functors from the orbit category

O_{frakturF} G

to the category of abelian groups and

{gd}_{frakturF} G

denotes the geometric Bredon dimension, that is, the minimal dimension of a model for the classifying space for proper actions of

G

.…”

Section: Introductionmentioning

confidence: 99%

“…The Gorenstein cohomological dimension of G over Z enjoys several useful properties and is proposed in [4] to serve as an algebraic invariant whose finiteness characterizes the groups which admit a finite-dimensional model for the classifying space for proper actions. It is related to many other numerical invariants that are studied in cohomological group theory: As shown in [15,24,26], for any group G we have a chain of inequalities…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Gorenstein dimension and group cohomology with group ring coefficients

Emmanouil

Talelli

2018

J. London Math. Soc.

View full text Add to dashboard Cite

The Gorenstein cohomological dimension of a group G generalizes the ordinary cohomological dimension of G, in the sense that the two invariants coincide when the latter one is finite. In this paper, we show that the Gorenstein cohomological dimension GcdkG of G over a commutative ring k shares many properties with the cohomological dimension of G over k. For example, if k has finite global dimension, the finiteness of GcdkG implies that GpdkGM is finite for any kG‐module M. Other properties concern the dependence of the Gorenstein cohomological dimension upon the coefficient ring, its behavior with respect to subgroups, the subadditivity with respect to extensions and the formula for the dimension of an ascending union. We also prove that if k has finite global dimension then the finiteness of GcdkG gives a criterion for the conditions cdkG<∞ and hdkG<∞ to be equivalent. Finally, we show that if k is a countable ring and G is a countable group of finite Gorenstein cohomological dimension over k, then GcdkG=supfalse{n:Hn(G,kG)≠0false}. The latter two results are special cases for group algebras of certain assertions that are valid over more general rings.

show abstract

“…The Gorenstein cohomological dimension of

G

over

double-struckZ

G

we have a chain of inequalities

{Gcd}_{double-struckZ} G ⩽ F {cd}_{double-struckZ} G ⩽ {cd}_{frakturF} G ⩽ {gd}_{frakturF} G ⩽ max false{3, {cd}_{F} G false} .

Here,

F {cd}_{double-struckZ} G

is the so‐called

frakturF

‐cohomological dimension of

G

over

double-struckZ

, which is defined in terms of relative homological algebra, relative to the class

frakturF

of finite subgroups of

G

. The Bredon cohomological dimension

{cd}_{frakturF} G

G

is defined in terms of the category of all contravariant functors from the orbit category

O_{frakturF} G

to the category of abelian groups and

{gd}_{frakturF} G

denotes the geometric Bredon dimension, that is, the minimal dimension of a model for the classifying space for proper actions of

G

.…”

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Gorenstein dimension and group cohomology with group ring coefficients

Emmanouil

Talelli

2018

J. London Math. Soc.

View full text Add to dashboard Cite

show abstract

“…If a finitely generated countable group G has an embedding G × G ֒→ G (e.g. the finitely presented "Thompson's group F ") then it has infinite rational cohomological dimension [22], so the contractibility hypothesis already excludes some "reasonable" countable groups from being autoequivalence groups.…”

Section: Stability Conditionsmentioning

confidence: 99%

Stability conditions in symplectic topology

Smith¹

2017

Preprint

View full text Add to dashboard Cite

show abstract

“…It also has jump cohomology of height one over Q (see [13,Theorem 4.11]). Yet this group has jump cohomology of height zero.…”

Section: Jump (Co)homologymentioning

confidence: 99%

Action-induced nested classes of groups and jump (co)homology

Petrosyan

2013

Forum Mathematicum

View full text Add to dashboard Cite

Using fixed-point-free group actions, we set up a scheme to define nested classes of groups indexed over ordinals. Restricting to cellular actions on CW-complexes, we find new classes of groups as well as new characterizations for some well-known classes. We extend some of the properties of the cohomological dimension of a group to groups with jump (co)homology and study their implications to cellular actions on finite dimensional CW-complexes that satisfy a quite general homological condition.

show abstract

Cohomological invariants and the classifying space for proper actions

Cited by 14 publications

References 23 publications

Gorenstein dimension and group cohomology with group ring coefficients

Gorenstein dimension and group cohomology with group ring coefficients

Stability conditions in symplectic topology

Action-induced nested classes of groups and jump (co)homology

Contact Info

Product

Resources

About