José Luis Cisneros-Molina

The η-Invariant of Twisted Dirac Operators of S3/Γ

The aim of this paper is to compute the η and ξ~-invariants for the Dirac operator of the quotient of the sphere S3 by a finite subgroup, twisted by a representation of its fundamental group.

An invariant of 2 by 2 matrices

Let W be the space of 2×2 matrices over a field K. Let f be any linear function on W that kills scalar matrices. Let A ∈ W and define fk(A) = f(Ak). Then the quantity fk+1(A)/f(A) is invariant under conjugation and moreover fk+1(A)/f(A) = traceS kA, where SkA is the k-th symmetric power of A, that is, the matrix giving the action of A on homogeneous polynomials of degree k.

Polynomials in Control Theory Parametrized by Their Roots

The aim of this paper is to introduce the space of roots to study the topological properties of the spaces of polynomials. Instead of identifying a monic complex polynomial with the vector of its coefficients, we identify it with the set of its roots. Vietes map gives a homeomorphism between the space of roots and the space of coefficients and it gives an explicit formula to relate both spaces. Using this viewpoint we establish that the space of monic (Schur or Hurwitz) aperiodic polynomials is contractible. Additionally we obtain a Boundary Theorem.

Singularities II: Geometric and Topological Aspects

In this paper we study the Milnor fibrations associated to real analytic map germs ψ : (R, 0) → (R, 0) with isolated critical point at 0 ∈ R. The main result relates the existence of called Strong Milnor fibrations with a transversality condition of a convenient family of analytic varieties with isolated critical points at the origin 0 ∈ R, obtained by projecting the map germ ψ in the family L −θ of all lines through the origin in the plane R .

On the topology of real analytic maps

We study the topology of the fibers of real analytic maps ℝn → ℝp, n > p, in a neighborhood of a critical point. We first prove that every real analytic map-germ f : ℝn → ℝp, p ≥ 1, with arbitrary critical set, has a Milnor–Le type fibration away from the discriminant. Now assume also that f has the Thom af-property, and its zero-locus has positive dimension. Also consider another real analytic map-germ g : ℝn → ℝk with an isolated critical point at the origin. We have Milnor–Le type fibrations for f and for (f, g) : ℝn → ℝp+k, and we prove for these the analogous of the classical Le–Greuel formula, expressing the difference of the Euler characteristics of the fibers Ff and Ff,g in terms of an invariant associated to these maps. This invariant can be expressed in various ways: as the index of the gradient vector field of a map

Classification of Isolated Polar Weighted Homogeneous Singularities

\tilde{g}

Invariants of hyperbolic 3-manifolds in relative group homology

on Ff associated to g; as the number of critical points of

Singularities I: Algebraic and Analytic Aspects

\tilde{g}

THE BLOCH INVARIANT AS A CHARACTERISTIC CLASS IN B(SL2(C);T)

on Ff; or in terms of polar multiplicities. When p = 1 and k = 1, this invariant can also be expre...

Relative group (co)homology theories with coefficients and the comparison homomorphism

Polar weighted homogeneous polynomials are real analytic maps which generalize complex weighted homogeneous polynomials. In this article we give classes of mixed polynomials in three variables which generalize Orlik and Wagreich classes of complex weighted homogeneous polynomials. We give explicit conditions for this classes to be polar weighted homogeneous polynomials with isolated critical point. We prove that under small perturbation of their coe_cients they remain with isolated critical point and the diffeomorphism type of their link does not change.