We define for arbitrary modules over a finite von Neumann algebra A a dimension taking values in [0,∞] which extends the classical notion of von Neumann dimension for finitely generated projective A-modules and inherits all its useful properties such as Additivity, Cofinality and Continuity. This allows to define L2-Betti numbers for arbitrary topological spaces with an action of a discrete gro...

Let R be a real closed field. The problem of obtaining tight bounds on the Betti numbers of semi-algebraic subsets of R in terms of the number and degrees of the defining polynomials has been an important problem in real algebraic geometry with the first results due to Olĕınik and Petrovskĭı, Thom and Milnor. These bounds are all exponential in the number of variables k. Motivated by several ap...

We generalize work of Lascoux and Jo zeeak-Pragacz-Weyman on Betti numbers for minimal free resolutions of ideals generated by 2 2 minors of generic matrices and generic symmetric matrices, respectively. Quotients of polynomial rings by these ideals are the classical Segre and quadratic Veronese subalgebras, and we compute the analogous Betti numbers for some natural modules over these Segre an...

Using graph representations a new class of computable topological invariants associated with a tame real or angle valued map were recently introduced, providing a theory which can be viewed as an alternative to MorseNovicov theory for real or angle valued Morse maps. The invariants are ”barcodes” and ”Jordan cells”. From them one can derive all familiar topological invariants which can be deriv...

Let R be a real closed field. We prove that for any fixed d, the equivariant rational cohomology groups of closed symmetric semi-algebraic subsets of Rk defined by polynomials of degrees bounded by d vanishes in dimensions d and larger. This vanishing result is tight. Using a new geometric approach we also prove an upper bound of dO(d)sdkbd/2c−1 on the equivariant Betti numbers of closed symmet...

We introduce a notion of complexity complex ℓ \ell -adic sheaves on quasi-projective variety and prove that the six operation...

This paper produces a recursive formula of the Betti numbers of certain Stanley-Reisner ideals (graph ideals associated to forests). This gives a purely combinatorial definition of the projective dimension of these ideals, which turns out to be a new numerical invariant of forests. Finally, we propose a possible extension of this invariant to general graphs. 0. Introduction Throughout this pape...

We prove that the Betti numbers of simplicial complexes of bounded vertex degrees are testable in constant time.

A generalization of the formula of Fine and Rao for the ranks of the intersection homology groups of a complex algebraic variety is given. The proof uses geometric properties of intersection homology and mixed Hodge theory. The middle-perversity intersection homology with integral coefficients of a compact complex n-dimensional algebraic variety X with isolated singularities is well known to be...