The Euler characteristic of the Whitehead automorphism group of a free product

A combinatorial summation identity over the lattice of labelled hypertrees is established that allows one to gain concrete information on the Euler characteristics of various automorphism groups of free products of groups. In particular, we establish formulae for the Euler characteristics of: the group of Whitehead automorphisms when the are of finite homological type; and when the are finite; and the palindromic automorphism groups of finite rank free groups.

     

The equivariant Euler characteristic

We describe an equivariant version of the Euler characteristic in order to extend to the equivariant case classical results relating the Euler characteristic to vector field (Reinhart) bordism of smooth manifolds and controllable cut-and-paste equivalence. We show that the nonequivariant results continue to hold for an arbitrary finite ambient group G, both in the oriented and unoriented cases, and thereby extend work on this subject begun by several authors. We use a new definition of equivariant orientation in terms of a categorical notion of ‘groupoid representations’.     

The Euler characteristic and valuations on MV-algebras

Every finitely presented MV-algebra A has a unique idempotent valuation E assigning value 1 to every basic element of A. For each a ∈ A, E(a) turns out to coincide with the Euler characteristic of the open set of maximal ideals m of A such that a/m is nonzero.     

Euler characteristic of Fredholm quasicomplexes

Quasicomplexes are usually understood as small (in some sense) perturbations of complexes. Of interest are not only perturbations within the category of complexes but also those going beyond this category. A sequence perturbed in this way is no longer a complex, and so it bears no cohomology. We show how to introduce the Euler characteristic for small perturbations of Fredholm complexes.     

The Euler characteristic is stable under compact perturbations

We prove in the general case the stability under compact perturbations of the index (i.e. the Euler characteristic) of a Fredholm complex of Banach spaces. In particular, we obtain the corresponding stability property for Fredholm multioperators. These results are the consequence of a similar statement, concerning more general objects called Fredholm pairs.

     

Sullivan's local Euler characteristic theorem

Using a certain cell decomposition of a closed neighborhood of a point a in a real analytic set A and the orientability modulo 2 of A ([1,3.7] or [5,7.3]), we obtain a short proof, by counting cells, of D. Sullivan's theorem ([9]) that X(A,A {a})) is odd.Research partially supported by NSF Grant GP29321.     

The vanishing Euler characteristic of an isolated determinantal singularity

Let (X, 0) be a complex analytic isolated determinantal singularity. We will define the vanishing Euler characteristic of (X, 0) and the Milnor number of a holomorphic function germ with an isolated singularity on X, f: (X, 0) → ?.     

On Euler characteristic of equivariant sheaves

Let k be an algebraically closed field of characteristic p>0 and let ? be another prime number. Gabber and Looser proved that for any algebraic torus T over k and any perverse ?-adic sheaf on T the Euler characteristic is non-negative.We conjecture that the same result holds for any perverse sheaf on a reductive group G over k which is equivariant with respect to the adjoint action. We prove the conjecture when is obtained by Goresky-MacPherson extension from the set of regular semi-simple elements in G. From this we deduce that the conjecture holds for G of semi-simple rank 1.     

The homotopy type of the polyhedral product for shifted complexes

We prove a conjecture of Bahri, Bendersky, Cohen and Gitler: if K$K$ is a shifted simplicial complex on n$n$ vertices, _X1,…,_Xn$X_1,\dots ,X_n$ are pointed connected CW$CW$-complexes and C_Xi$C{X}_i$ is the cone on  _Xi$X_i$, then the polyhedral product determined by K$K$ and the pairs (C_Xi,_Xi)$(C{X}_i,X_i)$ is homotopy equivalent to a wedge of suspensions of smashes of the _Xi$X_i$’s. Earlier work of the authors dealt with the special case where each _Xi$X_i$ is a loop space. New techniques are introduced to prove the general case. These have the advantage of simplifying the earlier results and of being sufficiently general to show that the conjecture holds for a substantially larger class of simplicial complexes. We discuss connections between polyhedral products and toric topology, combinatorics, and classical homotopy theory.     

On polyhedra with extremal Euler characteristic

It is the main purpose of this paper to determine the extremal values of the Euler characteristic of unions of at most n polytopes, disregarding the dimension d of the space in which they lie.     

Normal Abelian subgroups and Euler characteristic

We generalize Rosset's theorem which states that the Euler characteristic of a group G of type FL_C vanishes if G contains a torsion free normal subgroup. In our case the subgroup is allowed to have torsion (but must also have elements of infinite order). Under similar conditions on the regular covering of a finite CW-complex X, it is shown that the Euler characteristic of X is 0; this includes the special case where X is nilpotent.     

A note on equivariant Euler characteristic

We give a new equivariant cohomological characterization of the equivariant Euler characteristic of aG-simplicial set as defined by Brown. This implies in particular that the equivariant Euler characteristic is aG-homotopy invariant.     

The complexity to compute the Euler characteristic of complex varieties

We extend one of the main results of Bürgisser and Cucker (http://www.arxiv.org/abs/cs/cs.CC/0312007), which asserts that the computation of the Euler characteristic of a semialgebraic set is complete in the counting complexity class ${\mathrm{FP}}_{\mathbb{R}}^{\mathrm{\#}{\mathrm{P}}_{\mathbb{R}}}$. The goal is to prove a similar result over $\mathbb{C}$: the computation of the Euler characteristic of an affine or projective complex variety is complete in the class ${\mathrm{FP}}_{\mathbb{C}}^{\mathrm{\#}{\mathrm{P}}_{\mathbb{C}}}$. To cite this article: P. Bürgisser et al., C. R. Acad. Sci. Paris, Ser. I 339 (2004).     

Bicovariograms and Euler characteristic of regular sets

We establish an expression of the Euler characteristic of a r‐regular planar set in function of some variographic quantities. The usual framework is relaxed to a regularity assumption, generalising existing local formulas for the Euler characteristic. We give also general bounds on the number of connected components of a measurable set of in terms of local quantities. These results are then combined to yield a new expression of the mean Euler characteristic of a random regular set, depending solely on the third order marginals for arbitrarily close arguments. We derive results for level sets of some moving average processes and for the boolean model with non‐connected polyrectangular grains in . Applications to excursions of smooth bivariate random fields are derived in the companion paper 25 , and applied for instance to Gaussian fields, generalising standard results.