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Homotopy equivalence of posets with a group action

J Thévenaz, P.J Webb
1991 Journal of combinatorial theory. Series A  
Then 4 is 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.  ... 
doi:10.1016/0097-3165(91)90030-k fatcat:bh3filkmzzcbrfv6lelyuelyfq

Equivariant homotopy of posets and some applications to subgroup lattices

Volkmar Welker
1995 Journal of combinatorial theory. Series A  
In this paper we consider the 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.  ... 
doi:10.1016/0097-3165(95)90107-8 fatcat:ad6dlfrtpzhipm3i2qgjtw5nxi

G-functors, G-posets and homotopy decompositions of G-spaces

Stefan Jackowski, Jolanta Słomińska
2001 Fundamenta Mathematicae  
For 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 −).  ... 
doi:10.4064/fm169-3-4 fatcat:m5iojedw6je55fx63kpbnmf4mi

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]

Jonathan Rubin
2020 arXiv   pre-print
We introduce new Elmendorf constructions for equivariant categories and 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  ... 
arXiv:2006.08876v1 fatcat:itiguwkqubhklgwnlotrqqlwea

An equivariant discrete model for complexified arrangement complements [article]

Emanuele Delucchi, Michael J. Falk
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.  ... 
arXiv:1305.0134v1 fatcat:vvgogwawlzcsvd62dg4jcbecna

Homotopy colimits - comparison lemmas for combinatorial applications

Volkmar Welker, Gnter M. Ziegler, Rade T. Zivaljevic
1999 Journal für die Reine und Angewandte Mathematik  
We provide 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.  ... 
doi:10.1515/crll.1999.035 fatcat:sximrlw5abfxxgi3uw4b36m4oe

On the subnormalizer of a p-subgroup

Carlo Casolo
1992 Journal of Pure and Applied Algebra  
., On the subnormalizer 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.  ... 
doi:10.1016/0022-4049(92)90139-7 fatcat:vsvnzvyl6nhxznjzojbcelelfy

Spherical posets from commuting elements

Cihan Okay
2018 Journal of group theroy  
We prove that the universal cover 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.  ... 
doi:10.1515/jgth-2018-0008 fatcat:yvwdxm5x3nhelkvsd4ecaw6qxq

Homology representations of unitary reflection groups [article]

Justin Koonin
2013 arXiv   pre-print
This paper continues the study 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.  ... 
arXiv:1303.5155v2 fatcat:xjtqe2w52nfghhudwut6hnhkfq

Propagating sharp group homology decompositions

Jesper Grodal, Stephen D. Smith
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 |.  ... 
doi:10.1016/j.aim.2005.01.006 fatcat:o3aik36puzb5dawe6owjsbl7ki

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

John Maginnis, Silvia Onofrei
2010 Journal of combinatorial theory. Series A  
We determine the nature 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.  ... 
doi:10.1016/j.jcta.2009.10.012 fatcat:nirv562rsrf5dpbmautmoz4yoy

Homotopy properties of the poset of nontrivial p-subgroups of a group

Daniel Quillen
1978 Advances in Mathematics  
If Z, is contractible for each x in X, then p,: Z -+ X is 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.  ... 
doi:10.1016/0001-8708(78)90058-0 fatcat:impq66ypbndphftnzc3grios3e

A nice acyclic matching on the nerve of the partition lattice [article]

Ralf Donau
2020 arXiv   pre-print
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≥ 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.  ... 
arXiv:1204.2693v4 fatcat:wranbfpgdfegld6wgi6nspvndu

Homotopy decompositions of orbit spaces and the Webb conjecture

Jolanta Słomińska
2001 Fundamenta Mathematicae  
We prove that if G is 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  ... 
doi:10.4064/fm169-2-2 fatcat:5kgzh3jrfffhrihrfrgwenrntu
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