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A mixed function is a real analytic map $f\colon \mathbb{C}^n \to \mathbb{C}$ in the complex variables $z_1,\dots,z_n$ and their conjugates $\bar{z}_1,\dots,\bar{z}_n$. In this article we define an integer valued index for vector fields $v$ with isolated singularity at $\mathbf{0}$ on real analytic varieties $V_f:=f^{-1}(0)$ defined by mixed functions $f$ with isolated critical point at $\mathbf{0}$. We call this index the mixed GSV-index and it generalizes the classical GSV-index defined by Gomez-Mont, Seade and Verjovsky, i.e., if the function $f$ is holomorphic, then the mixed GSV-index coincides with the GSV-index. Furthermore, the mixed GSV-index is a lifting to $\mathbb{Z}$ of the $\mathbb{Z}_2$-valued real GSV-index defined by Aguilar, Seade and Verjovsky. As applications we prove that the mixed GSV-index is equal to the Poincar\'e-Hopf index of $v$ on a Milnor fiber. If $f$ also satisfies the strong Milnor condition, i.e., for every $\epsilon>0$ (small enough) the map $\frac{f}{\|f\|}\colon \mathbb{S}_\epsilon \setminus L_f \to \mathbb{S}^1$ is a fiber bundle, we prove that the mixed GSV-index is equal to the curvatura integra of $f$ defined by Cisneros-Molina, Grulha and Seade based on the curvatura integra defined by Kervaire.

Let $L$ be a compact oriented $3$-manifold and $\rho\colon\pi_1(L)\to \mathrm{GL}(n,\mathbb{C})$ a representation. Evaluating the Cheeger-Chern-Simons class $\widehat{c}_{\rho,k}\in H^{2k-1}(L;\mathbb{C}/\mathbb{Z})$ of $\rho$ at $\nu\in H_{2k-1}(L;\mathbb{Z})$ we get characteristic numbers that we call the $k$-th CCS-numbers of $\rho$. We prove that if $\rho$ is a topologically trivial representation, the 2-nd CCS-number $\widehat{c}_{\rho,2}([L])$ of the fundamental class $[L]$ of $L$ is given by the invariant $\tilde{\xi}_\rho(D)$ of the Dirac operator $D$ of $L$ twisted by $\rho$ defined by Atiyah, Patodi and Singer. If $L$ is a rational homology sphere, we also give a formula for $\widehat{c}_{\rho,2}([L])$ of any representation $\rho$ in terms of $\tilde{\xi}(D)$. Given a topologically trivial representation $\rho\colon\pi_1(L)\to\mathrm{GL}(n,\mathbb{C})$ we construct an element $\langle L,\rho\rangle$ in the $3$-rd algebraic K-theory group $K_3(\mathbb{C})$ of the complex numbers. For a finite subgroup $\Gamma$ of $\mathrm{SU(2)}$ and its irreducible representations, we compute the 1-st and 2-nd CCS-numbers. With this, we recover the spectrum of all rational double point singularities. Motivated by this result, we define the topological spectrum of rational surface singularities and Gorenstein singularities. Given a normal surface singularity $(X,x)$ with link a rational homology sphere $L$, we show how to compute the invariant $\tilde{\xi}_\rho(D)$ for the Dirac operator of $L$ using either a resolution or a smoothing of $(X,x)$.

In [22] Milnor proved that a real analytic map $f\colon (R^n,0) \to (R^p,0)$, where $n \geq p$, with an isolated critical point at the origin has a fibration on the tube $f|\colon B_\epsilon^n \cap f^{-1}(S_\delta^{p-1}) \to S_\delta^{p-1}$. 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 $\varphi\colon S_\epsilon^{n-1} \setminus f^{-1}(0) \to S^{p-1}$, but the projection is not necessarily given by $f/ \|f\|$ as in the complex case. In the case $f$ has isolated critical value, in [9] it was proved that if the fibres inside a small tube are transverse to the sphere $S_\epsilon$, then it has a fibration on the tube. Also in [9], the concept of $d$-regularity was defined, it turns out that $f$ is $d$-regular if and only if the map $f/|f\|\colon S_\epsilon^{n-1} \setminus f^{-1}(0) \to S^{p-1}$ is a fibre bundle equivalent to the one on the tube. In this article, we prove the corresponding facts in a more general setting: if a locally surjective map $f$ has a linear discriminant $\Delta$ and a fibration on the tube $f|\colon B_\epsilon^n \cap f^{-1}(S_\delta^{p-1} \setminus \Delta) \to S_\delta^{p-1} \setminus \Delta$, then $f$ is $d$-regular if and only if the map $f/ \|f\|\colon S_\epsilon^{n-1} \setminus f^{-1}(\Delta) \to S^{p-1} \setminus \mathcal{A}$ (with $\mathcal{A}$ the radial projection of $\Delta$ on $S^{p-1}$) is a fibre bundle equivalent to the one on the tube. We do this by constructing a vector field $\tilde{w}$ 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 $\tilde{w}$., Comment: 23 pages. This is a corrected version of the article published in Internat. J. Math., 30(14):1950078, 1-25, 2019. https://www.worldscientific.com/doi/abs/10.1142/S0129167X19500782

We prove fibration theorems \`a la Milnor for differentiable real maps with non isolated critical values. We study the situation for maps with linear discriminant, and prove that the concept of d-regularity is the key point for the existence of a Milnor fibration on the sphere. We also explain how one can modify the target space by homeomorphisms to linearize a general discriminant. Whenever the composed map is d-regular one has fibration on the sphere. Plenty of examples are discussed along the text.

The Image-Computing Spectral Sequence computes the homology of the image of a finite map from the alternating homology of the multiple point spaces of the map. A related spectral sequence was obtained by Gabrielov, Vorobjob and Zell which computes the homology of the image of a closed map from the homology of $k$-fold fibred products of the map. We give new proofs of these results, in case the map can be triangulated. Thanks to work of Hardt, this holds for a very wide range of maps, and in particular for most of the finite maps of interest in singularity theory. The proof seems conceptually simpler and more canonical than earlier proofs., Comment: 20 pages

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 (co)homology theories 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-Nev\'arez from Takasu's theory to Adamson's one. 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 L\"uck-Wiermann construction to introduce a long exact sequence for Adamson (co)homology. Finally, we provide some examples of explicit computations for the comparison homomorphism., Comment: Shortened version. This version was accepted for publication in Quaestiones Mathematicae

Let $M$ be a complete oriented hyperbolic $3$--manifold of finite volume. Using classifying spaces for families of subgroups we construct a class $\beta_P(M)$ in the Adamson relative homology group $H_3([PSL_2(\mathbb{C}):\bar{P}];\mathbb{Z})$, where $\bar{P}$ is the subgroup of parabolic transformations which fix $\infty$ in the Riemann sphere. We also prove that the classes $F(M)$ in the Takasu relative homology groups $H_3(PSL_2(\mathbb{C}),\bar{P};\mathbb{Z})$ constructed by Zickert, which are not well-defined and depend of a choice of decorations by horospheres, are all mapped to $\beta_P(M)$ via a canonical comparison homomorphism $H_3(PSL_2(\mathbb{C}),\bar{P};\mathbb{Z})\to H_3([PSL_2(\mathbb{C}):\bar{P}];\mathbb{Z})$. To do this, we simplify the construction of the classes $F(M)$ using a simpler complex which computes $H_3(PSL_2(\mathbb{C}),\bar{P};\mathbb{Z})$, getting a simple simplicial formula for $F(M)$, which in turn gives a simpler and more efficient formula to compute the volume and Chern--Simons invariant than the one given by Zickert. The constructions can be extended for any boundary-parabolic $PSL_2(\mathbb{C})$-representation., Comment: 47 pages, 1 figure, PDFLaTeX. The proof of Proposition 3.20 was not correct (and probably also the statement). Results on Sections 2 and 3 were published in a separate article and summarized in Section 2 of version 2. Some parts were rewritten

Let $X$ be an analytic subset of an open neighbourhood $U$ of the origin $\underline{0}$ in $\mathbb{C}^n$. Let $f\colon (X,\underline{0}) \to (\mathbb{C},0)$ be holomorphic and set $V =f^{-1}(0)$. Let $\B_\epsilon$ be a ball in $U$ of sufficiently small radius $\epsilon>0$, centred at $\underline{0}\in\mathbb{C}^n$. We show that $f$ has an associated canonical pencil of real analytic hypersurfaces $X_\theta$, with axis $V$, which leads to a fibration $\Phi$ of the whole space $(X \cap \mathbb{B}_\epsilon) \setminus V$ over $\mathbb{S}^1 $. Its restriction to $(X \cap \mathbb{S}_\epsilon) \setminus V$ is the usual Milnor fibration $\phi = \frac{f}{|f|}$, while its restriction to the Milnor tube $f^{-1}(\partial \D_\eta) \cap \mathbb{B}_\epsilon$ is the Milnor-L\^e fibration of $f$. Each element of the pencil $X_\theta$ meets transversally the boundary sphere $\mathbb{S}_\epsilon = \partial \B_\epsilon$, and the intersection is the union of the link of $f$ and two homeomorphic fibers of $\phi$ over antipodal points in the circle. Furthermore, the space ${\tilde X}$ obtained by the real blow up of the ideal $(Re(f), Im(f))$ is a fibre bundle over $\mathbb{R} \mathbb{P}^1$ with the $X_\theta$ as fibres. These constructions work also, to some extent, for real analytic map-germs, and give us a clear picture of the differences, concerning Milnor fibrations, between real and complex analytic singularities., Comment: 37 pages, LaTeX; slightly modified title and abstract, rewrote introduction, reorganized parts of the paper and references added; some errors have been fixed and some improved results added; some lemmas added and a proof extended. To appear in Advances in Mathematics