Virtual Betti Numbers Research Articles

Virtual Betti numbers of mapping tori of 3-manifolds

Given a reducible $3$-manifold $M$ with an aspherical summand in its prime decomposition and a homeomorphism $f\colon M\to M$, we construct a map of degree one from a finite cover of $M\rtimes_f S^1$ to a mapping torus of a certain aspherical $3$-manifold. We deduce that $M\rtimes_f S^1$ has virtually infinite first Betti number, except when all aspherical summands of $M$ are virtual $T^2$-bundles. This verifies all cases of a conjecture of T.-J. Li and Y. Ni, that any mapping torus of a reducible $3$-manifold $M$ not covered by $S^2\times S^1$ has virtually infinite first Betti number, except when $M$ is virtually $(\#_n T^2\rtimes S^1)\#(\#_mS^2\times S^1)$. Li-Ni's conjecture was recently confirmed by Ni with a group theoretic result, namely, by showing that there exists a $\pi_1$-surjection from a finite cover of any mapping torus of a reducible $3$-manifold to a certain mapping torus of $\#_m S^2\times S^1$ and using the fact that free-by-cyclic groups are large when the free group is generated by more than one element.

Equivariant weight filtration for real algebraic varieties with action

We show the existence of an equivariant weight filtration on the equivariant homology of real algebraic varieties equipped with a finite group action, by applying group homology onto the weight complex of McCrory and Parusi\'nski. The group action enriches the information contained in the induced equivariant weight spectral sequence from which we can not extract finite additivity directly if the group is of even order. Still, we manage to construct additive invariants from its elementary bricks, and prove that they are induced from the very equivariant geometry of real algebraic varieties with action, and that they coincide with Fichou's equivariant virtual Betti numbers in some cases. In the case of the two-elements group, we recover these through the use of globally invariant chains and the equivariant version of Guill{\'e}n and Navarro Aznar's extension criterion.

Cohomology and products of real weight filtrations

We associate to each algebraic variety defined over $\mathbb{R}$ a filtered cochain complex, which computes the cohomology with compact supports and $\mathbb{Z}\_2$-coefficients of the set of its real points. This filtered complex is additive for closed inclusions and acyclic for resolution of singularities, and is unique up to filtered quasi-isomorphism. It is represented by the dual filtration of the geometric filtration on semialgebraic chains with closed supports defined by McCrory and Parusi\'nski, and leads to a spectral sequence which computes the weight filtration on cohomology with compact supports. This spectral sequence is a natural invariant which contains the additive virtual Betti numbers. We then show that the cross product of two varieties allows us to compare the product of their respective weight complexes and spectral sequences with those of their product, and prove that the cup and cap products in cohomology and homology are filtered with respect to the real weight filtrations.

Virtual Betti numbers and the symplectic Kodaira dimension of fibered $4$-manifolds

Virtual Betti numbers and the symplectic Kodaira dimension of fibered $4$-manifolds

Virtual Betti numbers and virtual symplecticity of 4-dimensional mapping tori

In this note, we compute the virtual first Betti numbers of 4-manifolds fibering over $$S^1$$ with prime fiber. As an application, we show that if such a manifold is symplectic with nonpositive Kodaira dimension, then the fiber itself is a sphere or torus bundle over $$S^1$$ . In a different direction, we prove that if the 3-dimensional fiber of such a 4-manifold is virtually fibered then the 4-manifold is virtually symplectic unless its virtual first Betti number is 1.

Motivic Milnor Fibers over Complete Intersection Varieties and their Virtual Betti Numbers

We study the Jordan normal forms of the local and global monodromies over complete intersection subvarieties of by using the theory of motivic Milnor fibers. The results will be explicitly described by the mixed volumes of the faces of Newton polyhedrons.

Virtual Betti numbers of compact locally symmetric spaces

We show that the virtual Betti number of a compact locally symmetric space with arithmetic fundamental group is either the Betti number of the compact dual or else is infinite.

Motivic invariants of arc-symmetric sets and blow-Nash equivalence

We define invariants of the blow-Nash equivalence of Nash function germs, in a similar way to the motivic zeta functions of Denef and Loeser. As a key ingredient, we extend the virtual Betti numbers, which were known for real algebraic sets, to a generalized Euler characteristic for projective constructible arc-symmetric sets. Actually we prove more: the virtual Betti numbers are not only algebraic invariants, but also Nash invariants of arc-symmetric sets. Our zeta functions enable one to distinguish the blow-Nash equivalence classes of Brieskorn polynomials of two variables. We prove moreover that there are no moduli for the blow-Nash equivalence in the case of an algebraic family with isolated singularities.

Virtual Betti numbers of real algebraic varieties

We show that for all i⩾0 the i-th mod 2 Betti number of compact nonsingular real algebraic varieties has a unique extension to a virtual Betti number β i defined for all real algebraic varieties, such that if Y is a closed subvariety of X then β i ( X)= β i ( X⧹ Y)+ β i ( Y). We show by example that there is no natural weight filtration on the Z 2 -cohomology of real algebraic varieties with compact supports such that the virtual Betti numbers are the weighted Euler characteristics. To cite this article: C. McCrory, A. Parusiński, C. R. Acad. Sci. Paris, Ser. I 336 (2003).

Cycles géodésiques transverses dans les variétés hyperboliques

Let M be a hyperbolic manifold, with \( \pi_1M \) finitely generated. Let c1 and c2 be two transverse geodesic cycles with dim(c1) + dim(c2) = dim M and \( c_1 \cap c_2 \neq \emptyset \). In this paper, following ideas of [MR], we prove (Theorem 6) that c1 and c2 lifts to a finite cover of M as two submanifolds F1 and F2 with¶¶\( [F_1][F_2] \neq 0 \).¶ This theorem implies in particular that the compact hyperbolic manifolds constructed by Gromov and Piatetski-Shapiro in [GP] have nontrivial virtual Betti numbers.

Sur l'homologie de cycles géodésiques dans des variétés hyperboliques compactes

The notion of an l-geodesic cycle in a compact hyperbolic n-manifold M generalises, in dimension l, the one of a closed geodesic. In this Note we show that when l ≥n/2, such a cycle lifts to a finite cover of M as an embedded totally geodesic submanifold non zero homologous. It enables us to prove that the compact hyperbolic manifolds constructed by Gromov and Piateski-Shapiro (see [4]) have infinite virtual Betti numbers and to give a new proof of the same fact for the compact arithmetic hyperbolic manifolds constructed by Borel in [3].