2,430 research outputs found
A combinatorial Lefschetz fixed-point formula
AbstractLet K be any (finite) simplicial complex, and Kβ² a subdivision of K. Let Ο: Kβ² β K be a simplicial map, and, for all j β©Ύ 0, let Οj denote the algebraical number of j-simplices G of Kβ² such that G β Ο(G). From Hopf's alternating trace formula it follows that Ο0 β Ο1 + Ο2 β β¦ = L(Ο), the Lefschetz number of the simplicial map Ο: X β X. Here X denotes the space of |K| (or |Kβ²|). A purely combinatorial proof of the case K = a closed simplex (now L(Ο) = 1) is given, thus solving a problem posed by Ky Fan in 1978
A Brouwer fixed point theorem for graph endomorphisms
We prove a Lefschetz formula for general simple graphs which equates the
Lefschetz number L(T) of an endomorphism T with the sum of the degrees i(x) of
simplices in G which are fixed by T. The degree i(x) of x with respect to T is
defined as a graded sign of the permutation T induces on the simplex x
multiplied by -1 if the dimension of x is odd. The Lefschetz number is defined
as in the continuum as the super trace of T induced on cohomology. In the
special case where T is the identity, the formula becomes the Euler-Poincare
formula equating combinatorial and cohomological Euler characteristic. The
theorem assures in general that if L(T) is nonzero, then T has a fixed clique.
A special case is a discrete Brouwer fixed point theorem for graphs: if T is a
graph endomorphism of a connected graph G, which is star-shaped in the sense
that only the zeroth cohomology group is nontrivial, like for connected trees
or triangularizations of star shaped Euclidean domains, then there is clique x
which is fixed by T. Unlike in the continuum, the fixed point theorem proven
here looks for fixed cliques, complete subgraphs which play now the role of
"points" in the graph. Fixed points can so be vertices, edges, fixed triangles
etc. If A denotes the automorphism group of a graph, we also look at the
average Lefschetz number L(G) which is the average of L(T) over A. We prove
that this is the Euler characteristic of the graph G/A and especially an
integer. We also show that as a consequence of the Lefschetz formula, the zeta
function zeta(T,z) is a product of two dynamical zeta functions and therefore
has an analytic continuation as a rational function which is explicitly given
by a product formula involving only the dimension and the signature of prime
orbits of simplices in G.Comment: 24 pages, 6 figure
A Hadwiger Theorem for Simplicial Maps
We define the notion of valuation on simplicial maps between geometric
realizations of simplicial complexes in . Valuations on
simplicial maps are analogous to valuations on sets. In particular, we define
the Lefschetz volumes, which are analogous to the intrinsic volumes of subsets
of . Our definition not only provides a generalization of the
Lefschetz number, but also yields a Hadwiger-style classification theorem for
all such valuations.Comment: 11 pages, 3 figure
Remarks on the combinatorial intersection cohomology of fans
We review the theory of combinatorial intersection cohomology of fans
developed by Barthel-Brasselet-Fieseler-Kaup, Bressler-Lunts, and Karu. This
theory gives a substitute for the intersection cohomology of toric varieties
which has all the expected formal properties but makes sense even for
non-rational fans, which do not define a toric variety. As a result, a number
of interesting results on the toric and polynomials have been extended
from rational polytopes to general polytopes. We present explicit complexes
computing the combinatorial IH in degrees one and two; the degree two complex
gives the rigidity complex previously used by Kalai to study . We present
several new results which follow from these methods, as well as previously
unpublished proofs of Kalai that implies and
.Comment: 34 pages. Typos fixed; final version, to appear in Pure and Applied
Math Quarterl
Coxeter group actions on the complement of hyperplanes and special involutions
We consider both standard and twisted action of a (real) Coxeter group G on
the complement M_G to the complexified reflection hyperplanes by combining the
reflections with complex conjugation. We introduce a natural geometric class of
special involutions in G and give explicit formulae which describe both actions
on the total cohomology H(M_G,C) in terms of these involutions. As a corollary
we prove that the corresponding twisted representation is regular only for the
symmetric group S_n, the Weyl groups of type D_{2m+1}, E_6 and dihedral groups
I_2 (2k+1) and that the standard action has no anti-invariants. We discuss also
the relations with the cohomology of generalised braid groups.Comment: 11 page
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