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In the proof of Theorem 3.7 of the original paper Internat. J. Math.30(14) (2019) 1950078, 1–25, two inequalities are used that do not hold in general. In this note, we prove an extra propositions which allows us to give a proof of Theorem 3.7 without using the aforementioned inequalities. Hence, all the results in the original paper are valid. We have also posted a corrected version in the arXiv. [ABSTRACT FROM AUTHOR]

Let G be a group, let H be a subgroup of G and let Or(G) be the orbit category. In this paper we extend the definition of the relative group homology the- ories of the pair (G, H) defined by Adamson and Takasu to have coefficients in an Or(G)-module. There is a canonical comparison homomorphism defined by Cisneros- Molina and Arciniega-Nevrez from Takasu's theory to Adamson's theory. We give a necessary and sufficient condition on the subgroup H for which the comparison homomorphism is an isomorphism for all coefficients. We also use the Lück-Wiermann construction to introduce a long exact sequence for Adamson homology. Finally, we provide some examples of explicit computations for the comparison homomorphism. [ABSTRACT FROM AUTHOR]

We study real analytic isolated singularities of type f:(Cn+m+1,0)→(C,0) with f(z,w)=g(z)+∑i=1m+1hi(wi,w¯i), where g is holomorphic and each hi is a mixed polynomial, with z=(z1,⋯,zn) and w=(w1,⋯,wm+1). We construct a Lê's vanishing polyhedron for f, which describes the degeneration of its Milnor fiber Ff to the singular fiber. Then we prove that Ff is homotopy equivalent to the join Fg∗Fh1∗⋯∗Fhm+1, where Fg is the Milnor fiber of g and Fhi is the Milnor fiber of hi. This implies that Ff has the homotopy type of a bouquet of spheres Sn+m. So we can define the Milnor number μ(f) as the number of spheres in that bouquet, as in the complex setting. [ABSTRACT FROM AUTHOR]

In [J. Milnor, Singular points of complex hypersurfaces, Annals of Mathematics Studies, No. 61 (Princeton University Press, Princeton, NJ, 1968).] Milnor proved that a real analytic map f : (ℝ n , 0) → (ℝ p , 0) , where n ≥ p , with an isolated critical point at the origin has a fibration on the tube f | : 𝔹 𝜀 n ∩ f − 1 (𝕊 δ p − 1) → 𝕊 δ p − 1 . Constructing a vector field such that (1) it is transverse to the spheres, and (2) it is transverse to the tubes, he "inflates" the tube to the sphere, to get a fibration φ : 𝕊 𝜀 n − 1 ∖ f − 1 (0) → 𝕊 p − 1 , but the projection is not necessarily given by f / ∥ f ∥ as in the complex case. In the case f has isolated critical value, in [J. L. Cisneros-Molina, J. Seade and J. Snoussi, Milnor fibrations and d -regularity for real analytic singularities, Internat. J. Math.21(4) (2010) 419–434.] it was proved that if the fibers inside a small tube are transverse to the sphere 𝕊 𝜀 , then it has a fibration on the tube. Also in [J. L. Cisneros-Molina, J. Seade and J. Snoussi, Milnor fibrations and d -regularity for real analytic singularities, Internat. J. Math.21(4) (2010) 419–434.], the concept of d -regularity was defined, it turns out that f is d -regular if and only if the map f / ∥ f ∥ : 𝕊 𝜀 n − 1 ∖ f − 1 (0) → 𝕊 p − 1 is a fiber bundle equivalent to the one on the tube. In a more general setting, the corresponding facts are proved in [J. L. Cisneros-Molina, A. Menegon, J. Seade and J. Snoussi, Fibration theorems and d -regularity for differentiable maps-germs with non-isolated critical value, Preprint (2017).], showing that if a locally surjective map f has a linear discriminant Δ with isolated singularity and a fibration on the tube f | : 𝔹 𝜀 n ∩ f − 1 (𝕊 δ p − 1 ∖ Δ) → 𝕊 δ p − 1 ∖ Δ , then f is d -regular if and only if the map f / ∥ f ∥ : 𝕊 𝜀 n − 1 ∖ f − 1 (Δ) → 𝕊 p − 1 ∖ 𝒜 (with 𝒜 the radial projection of Δ on 𝕊 p − 1 ) is a fiber bundle equivalent to the one on the tube. In this paper, we generalize this result for an arbitrary linear discriminant by constructing a vector field w ̃ which inflates the tube to the sphere in a controlled way, it satisfies properties analogous to the vector field constructed by Milnor in the complex setting: besides satisfying (1) and (2) above, it also satisfies that f / ∥ f ∥ is constant on the integral curves of w ̃. [ABSTRACT FROM AUTHOR]

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–Lê 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–Lê type fibrations for f and for (f, g) : ℝn → ℝp+k, and we prove for these the analogous of the classical Lê–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 on Ff associated to g; as the number of critical points of on Ff; or in terms of polar multiplicities. When p = 1 and k = 1, this invariant can also be expressed algebraically, as the signature of a certain bilinear form. When the germs of f and (f, g) are both isolated complete intersection singularities, we exhibit an even deeper relation between the topology of the fibers Ff and Ff,g, and construct in this setting, an integer-valued invariant, that we call the curvatura integra that picks up the Euler characteristic of the fibers. This invariant, and its name, spring from Gauss' theorem, and its generalizations by Hopf and Kervaire, expressing the Euler characteristic of a manifold (with some conditions) as the degree of a certain map. [ABSTRACT FROM AUTHOR]

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. Viète's 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. [ABSTRACT FROM AUTHOR]