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Homotopy equivalence of posets with a group action

1991
*
Journal of combinatorial theory. Series A
*

Then 4 is

doi:10.1016/0097-3165(91)90030-k
fatcat:bh3filkmzzcbrfv6lelyuelyfq
*a*G-*homotopy**equivalence*. Here*a*G-*poset*means*a*partially ordered set together*with*an orderpreserving*action**of*G, and G, denotes the stabilizer*of*y. Moreover we define %Q= {ZEY I ZGY} y> ... These results deal*with*combinatorial topology, in particular*with*complexes associated*with**posets**of*subgroups*of**a*finite*group*. ...##
###
Equivariant homotopy of posets and some applications to subgroup lattices

1995
*
Journal of combinatorial theory. Series A
*

In this paper we consider the

doi:10.1016/0097-3165(95)90107-8
fatcat:ad6dlfrtpzhipm3i2qgjtw5nxi
*action**of**a*finite*group*G on the geometric realization ]CP[*of*the order complex CP*of**a**poset*P, on which*a**group*G acts as*a**group**of**poset*automorphisms. ... Moreover, we provide conditions which imply that the orbit space I CPI/G is*homotopy**equivalent*to the geometric realization*of*the order complex over the orbit*poset*P/G. ... If*a**group*G acts on the*poset*P as*a**group**of**poset*automorphisms (i.e., x <_ y ~ x g < yg*A*(V)°/V) are*homotopy**equivalent*. ...##
###
G-functors, G-posets and homotopy decompositions of G-spaces

2001
*
Fundamenta Mathematicae
*

For

doi:10.4064/fm169-3-4
fatcat:m5iojedw6je55fx63kpbnmf4mi
*a**group*G acting on*a**poset*W and an isotropy presheaf d : |W| is*a**homotopy**equivalence*. ... We describe*a*unifying approach to*a*variety*of**homotopy*decompositions*of*classifying spaces, mainly*of*finite*groups*. ... For any object G/K ∈ O (p) (G) the category I/(G/K) is*equivalent*to the category*of*K-orbits O (p) (K) and the functor on I/(G/K) we are interested in corresponds to the functor H * (EG × K −). ...##
###
Page 6012 of Mathematical Reviews Vol. , Issue 92k
[page]

1992
*
Mathematical Reviews
*

J. (1-MN)

*Homotopy**equivalence**of**posets**with**a**group**action*. J. Combin. Theory Ser.*A*56 (1991), no. 2, 173-181. ... Then g is*a*G-*homotopy**equivalence*. Here*a*G-*poset*means*a*partially ordered set together*with*an order-preserving*action**of*G, and G, denotes the stabilizer*of*y. ...##
###
Elmendorf constructions for G-categories and G-posets
[article]

2020
*
arXiv
*
pre-print

We introduce new Elmendorf constructions for equivariant categories and

arXiv:2006.08876v1
fatcat:itiguwkqubhklgwnlotrqqlwea
*posets*, and we prove that they are compatible*with*the classical topological one. ... Our constructions are more concrete than their model-categorical counterparts, and they give rise to new proofs*of*the Elmendorf theorems for equivariant categories and*posets*. ... Here,*a*weak*equivalence**of*categories or*posets*is*a*functor F : D → E such that BF is*a**homotopy**equivalence**of*spaces,*a*weak*equivalence**of*presheaves is*a*natural transformation λ : X ⇒ Y such that ...##
###
An equivariant discrete model for complexified arrangement complements
[article]

2013
*
arXiv
*
pre-print

*homotopy*-

*equivalent*to M under the identification

*of*Z_4

*with*+1, i, -1, -i, and |Q(M, e)| is

*homotopy*-

*equivalent*to the complement

*of*the decone

*of*

*A*relative to the hyperplane corresponding to e. ... 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). ... was

*a*postdoc there. ...

##
###
Homotopy colimits - comparison lemmas for combinatorial applications

1999
*
Journal für die Reine und Angewandte Mathematik
*

We provide

doi:10.1515/crll.1999.035
fatcat:sximrlw5abfxxgi3uw4b36m4oe
*a*"toolkit"*of*basic lemmas for the comparison*of**homotopy*types*of**homotopy*colimits*of*diagrams*of*spaces over small categories. ... We show how this toolkit can be used in quite different fields*of*applications. We demonstrate this*with*respect to 1. ... The referee's remarks, corrections and additions (see in particular Section 3.3) were extremely helpful, and led to*a*substantial improvement*of*the paper and its exposition. ...##
###
On the subnormalizer of a p-subgroup

1992
*
Journal of Pure and Applied Algebra
*

., On the subnormalizer

doi:10.1016/0022-4049(92)90139-7
fatcat:vsvnzvyl6nhxznjzojbcelelfy
*of**a*p-subgroup, Journal*of*Pure and Applied Algebra 77 (1992) 231-238. ... We say that g is*a**homotopy**equivalence*if I g] is*a**homotopy**equivalence*; in this case we say that the two*posets*X and Y are*homotopy**equivalent*. ... Then X<, is the union*of*two points, while gX,r = X,, is*homotopy**equivalent*to*a*circle. Mobius functions Let X be*a*finite*poset*. ...##
###
Spherical posets from commuting elements

2018
*
Journal of group theroy
*

We prove that the universal cover

doi:10.1515/jgth-2018-0008
fatcat:yvwdxm5x3nhelkvsd4ecaw6qxq
*of*its nerve is*homotopy**equivalent*to*a*wedge*of*r-spheres where {2r\geq 4} is the rank*of*its Frattini quotient. ... In this paper, we study the*homotopy*type*of*the partially ordered set*of*left cosets*of*abelian subgroups in an extraspecial p-*group*. ... Spherical*posets*also play*a*role in homological stability results. In [12] Vogtmann showed*a*stability result for the orthogonal*group*by considering the*action*on the*poset**of*singular subspaces. ...##
###
Homology representations of unitary reflection groups
[article]

2013
*
arXiv
*
pre-print

This paper continues the study

arXiv:1303.5155v2
fatcat:xjtqe2w52nfghhudwut6hnhkfq
*of*the*poset**of*eigenspaces*of*elements*of**a*unitary reflection*group*(for*a*fixed eigenvalue), which was commenced in [6] and [5]. ... The emphasis in this paper is on the representation theory*of*unitary reflection*groups*. The main tool is the theory*of**poset*extensions due to Segev and Webb ([16]). ... (iv) If P and Q are G-*posets*and φ : P → Q is*a**homotopy**equivalence**of**posets*and also*a*G-*poset*map, then φ is said to be*a*G-*homotopy**equivalence*. ...##
###
Propagating sharp group homology decompositions

2006
*
Advances in Mathematics
*

*A*collection C

*of*subgroups

*of*

*a*finite

*group*G can give rise to three different standard formulas for the cohomology

*of*G in terms

*of*either: the subgroups in C; or their centralizers; or their normalizers ... To do this, we exhibit some sufficient conditions on the

*poset*C which imply comparison results. ... If X is

*a*G-subposet

*of*

*a*G-

*poset*X and F is G-equivariant, then the retraction will be

*a*G-

*homotopy*deformation retraction, and |X | is G-

*homotopy*

*equivalent*to |X |. ...

##
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On fixed point sets of distinguished collections for groups of parabolic characteristic

2010
*
Journal of combinatorial theory. Series A
*

We determine the nature

doi:10.1016/j.jcta.2009.10.012
fatcat:nirv562rsrf5dpbmautmoz4yoy
*of*the fixed point sets*of**groups**of*order p, acting on complexes*of*distinguished p-subgroups (those p-subgroups containing p-central elements in their centers). ...*A**poset*map is*a*G-*homotopy**equivalence*if and only if the induced map on H -fixed points is*a**homotopy**equivalence*for all H G; see [18, 1.3] . Notation 2.1. ... Since Δ 5*A*is*homotopy**equivalent*to*a*building, Δ Q will be contractible for any 5-*group*Q*of*order at least 25 which contains an element*of*type 5*A*. ...##
###
Homotopy properties of the poset of nontrivial p-subgroups of a group

1978
*
Advances in Mathematics
*

If Z, is contractible for each x in X, then p,: Z -+ X is

doi:10.1016/0001-8708(78)90058-0
fatcat:impq66ypbndphftnzc3grios3e
*a**homotopy**equivalence*. The*poset*x\p, consists*of*all (x', y) in Z*with*x' 3 x. ... The*homotopy*property can be used to show*a*map f: X + Y is*a**homotopy**equivalence*(i.e., 1 f 1 is*a**homotopy**equivalence*) when there exists*a*map g: I' ---f -X*with*suitable properties (e.g., f, g are ... where T(Fn)*A*is the*poset**of*proper*A*-invariant subspaces*of*Fn. ...##
###
A nice acyclic matching on the nerve of the partition lattice
[article]

2020
*
arXiv
*
pre-print

The author has already proven that the space Δ(Π_n)/G is

arXiv:1204.2693v4
fatcat:wranbfpgdfegld6wgi6nspvndu
*homotopy**equivalent*to*a*wedge*of*spheres*of*dimension n-3 for all natural numbers n≥ 3 and all subgroups G⊂ S_1× S_n-1. ... We construct an S_1× 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. ... Furthermore the author would like to thank another anonymous reader*of**a*previous version*of*this paper. ...##
###
Homotopy decompositions of orbit spaces and the Webb conjecture

2001
*
Fundamenta Mathematicae
*

We prove that if G is

doi:10.4064/fm169-2-2
fatcat:5kgzh3jrfffhrihrfrgwenrntu
*a*compact Lie*group**with**a*non-trivial p-subgroup, then the orbit space (BA p (G))/G*of*the classifying space*of*the category associated to the G-*poset**A*p (G)*of*all non-trivial ... We also investigate some other equivariant*homotopy*and homology decompositions*of*X and prove that if G is*a*compact Lie*group**with**a*non-trivial p-subgroup, then the map EG × ... The category W [G] is*a*topological*poset**with*an*action**of*G defined by the*action**of*G on G/G w. and there is an equivariant isomorphism*of*topological G-*posets*F : W [G] → sd W such that F ([w.], [g ...
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