arXiv subjects

Haydée Aguilar-Cabrera

Publications and source records attributed to Haydée Aguilar-Cabrera.

3 recordsLinked to original sources

arXiv2013

Cyclic branched covers of knots as links of real isolated singularities

Given a real analytic function $f$ from $\mathbb{R}^4$ to $\mathbb{R}^2$ with isolated critical point at the origin, the link $L_f$ of the singularity is a real fibred knot in $\mathbb{S}^{3}$. From this singularities, we construct a family of real isolated suspension singularities from $\mathbb{R}^6$ to $\mathbb{R}^2$ such that its links are the total spaces of the $n$-branched cyclic coverings over the corresponding knots. In this way we obtain as links of singularities, $3$-manifolds that does not appear in the complex case, such as hyperbolic $3$-manifolds or the Hantzsche-Wendt manifold.

math.GT↗

arXiv2013

Open book decompositions of $\mathbb{S}^{5}$ and real singularities

In this article, we study the topology of the family of real analytic germs $F \colon (\mathbb{C}^3,0) \to (\mathbb{C},0)$ given by $F(x,y,z)=\bar{xy}(x^p+y^q)+z^r$ with $p,q,r \in \mathbb{N}$, $p,q,r \geq 2$ and $(p,q)=1$. Such a germ has isolated singularity at 0 and gives rise to a Milnor fibration $\frac{F}{|F|} \colon \mathbb{S}^{5} \setminus L_F \to \mathbb{S}^{1}$. We describe the link $L_F$ as a Seifert manifold and we show that it is always homeomorphic to the link of a complex singularity. However, we prove that in almost all the cases the open-book decomposition of $\mathbb{S}^{5}$ given by the Milnor fibration of $F$ cannot come from the Milnor fibration of a complex singularity in $\mathbb{C}^3$.

math.GT↗

arXiv2012

The topology of real suspension singularities of type $f \bar{g}+z^n$

In this article we study the topology of a family of real analytic germs $F \colon (\mathbb{C}^3,0) \to (\mathbb{C},0)$ with isolated critical point at 0, given by $F(x,y,z)=f(x,y)\bar{g(x,y)}+z^r$, where $f$ and $g$ are holomorphic, $r \in \mathbb{Z}^+$ and $r \geq 2$. We describe the link $L_F$ as a graph manifold using its natural open book decomposition, related to the Milnor fibration of the map-germ $f \bar{g}$ and the description of its monodromy as a quasi-periodic diffeomorphism through its Nielsen invariants. Furthermore, such a germ $F$ gives rise to a Milnor fibration $\frac{F}{|F|} \colon \mathbb{S}^5 \setminus L_F \to \mathbb{S}^1$. We present a join theorem, which allows us to describe the homotopy type of the Milnor fibre of $F$ and we show some cases where the open book decomposition of $\mathbb{S}^5$ given by the Milnor fibration of $F$ cannot come from the Milnor fibration of a complex singularity in $\mathbb{C}^3$.

math.GT↗