72 research outputs found
Combinatorics of Topological Posets:\ Homotopy complementation formulas
We show that the well known {\em homotopy complementation formula} of
Bj\"orner and Walker admits several closely related generalizations on
different classes of topological posets (lattices). The utility of this
technique is demonstrated on some classes of topological posets including the
Grassmannian and configuration posets, and
which were introduced and studied by V.~Vassiliev. Among other
applications we present a reasonably complete description, in terms of more
standard spaces, of homology types of configuration posets which
leads to a negative answer to a question of Vassilev raised at the workshop
``Geometric Combinatorics'' (MSRI, February 1997)
Order complexes of noncomplemented lattices are nonevasive
We reprove and generalize in a combinatorial way the result of A. Bj\"orner
[J.\ Comb.\ Th.\ A {\bf 30}, 1981, pp.~90--100, Theorem 3.3], that order
complexes of noncomplemented lattices are contractible, namely by showing that
these simplicial complexes are in fact nonevasive, in particular collapsible
Complexes of not -connected graphs
Complexes of (not) connected graphs, hypergraphs and their homology appear in
the construction of knot invariants given by V. Vassiliev. In this paper we
study the complexes of not -connected -hypergraphs on vertices. We
show that the complex of not -connected graphs has the homotopy type of a
wedge of spheres of dimension . This answers one of the
questions raised by Vassiliev in connection with knot invariants. For this case
the -action on the homology of the complex is also determined. For
complexes of not -connected -hypergraphs we provide a formula for the
generating function of the Euler characteristic, and we introduce certain
lattices of graphs that encode their topology. We also present partial results
for some other cases. In particular, we show that the complex of not
-connected graphs is Alexander dual to the complex of partial matchings
of the complete graph. For not -connected graphs we provide a formula
for the generating function of the Euler characteristic
Chains of modular elements and shellability
Let L be a lattice admitting a left-modular chain of length r, not
necessarily maximal. We show that if either L is graded or the chain is
modular, then the (r-2)-skeleton of L is vertex-decomposable (hence shellable).
This proves a conjecture of Hersh. Under certain circumstances, we can find
shellings of higher skeleta. For instance, if the left-modular chain consists
of every other element of some maximum length chain, then L itself is
shellable. We apply these results to give a new characterization of finite
solvable groups in terms of the topology of subgroup lattices.
Our main tool relaxes the conditions for an EL-labeling, allowing multiple
ascending chains as long as they are lexicographically before non-ascending
chains. We extend results from the theory of EL-shellable posets to such
labelings. The shellability of certain skeleta is one such result. Another is
that a poset with such a labeling is homotopy equivalent (by discrete Morse
theory) to a cell complex with cells in correspondence to weakly descending
chains.Comment: 20 pages, 1 figure; v2 has minor fixes; v3 corrects the technical
lemma in Section 4, and improves the exposition throughou
Homotopy colimits – comparison lemmas for combinatorial applications
Dieser Beitrag ist mit Zustimmung des Rechteinhabers aufgrund einer (DFG geförderten) Allianz- bzw. Nationallizenz frei zugänglich.This publication is with permission of the rights owner freely accessible due to an Alliance licence and a national licence (funded by the DFG, German Research Foundation) respectively.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. Björner's “Generalized Homotopy Complementation Formula” [5], 2. the topology of toric varieties, 3. the study of homotopy types of arrangements of subspaces, 4. the analysis of homotopy types of subgroup complexes
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