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Valentina Bais

Publications and source records attributed to Valentina Bais.

12 recordsLinked to original sources
arXiv2026

Cobordism groups of dihedral branched covers

For every integer $n \geq 1$, we compute the cobordism groups of dihedral $n$-fold branched covers of $S^3$ with oriented and non-oriented branching sets. We show that the groups are cyclic, generated by the $n$-fold connected dihedral covers of $(2,n)$-torus links. The isomorphism types of the groups are detected using explicit cobordism invariants defined in terms of the Seifert forms on the branching sets, generalizing Cappell-Shaneson characteristic knots associated to dihedral covers.

math.GT↗
arXiv2026

Branched coverings of simply connected $4$-manifolds

We show that, given $d \geq 4$ and two closed connected oriented PL $4$-manifolds $M$ and $N$ such that $N$ has a handle decomposition with no $1$- and $3$-handles, there exists a $d$-fold (simple) branched covering $p \colon M \rightarrow N$ if and only if there is an isometric embedding of lattices $d \cdot I_N \hookrightarrow I_M$. Here $I_N$ and $I_M$ respectively denote the intersection lattices of $N$ and $M$. In particular, we characterize the manifolds which are branched covers of the K3 surface.

math.GT↗
arXiv2025

On the detection of knotted spheres by their traces in high dimensions

For every $n \geq 4$, we demonstrate the existence of non-isotopic smooth $(n-2)$-knots in $S^n$ with diffeomorphic traces by generalising the RBG link construction to all dimensions. Conversely, we prove that for every $n \geq 4$, the unknot in $S^n$ is detected by the diffeomorphism type of its surgery and hence by its trace.

math.GT↗
arXiv2025

Branched covering representation of non-orientable $4$-manifolds

We show that every closed connected non-orientable PL $4$-manifold $X$ is a simple branched covering of $\RP^4$. We also show that $X$ is a simple branched covering of the twisted $S^3$-bundle $S^1 \simtimes S^3$ if and only if the first Stiefel--Whitney class $w_1(X)$ admits an integral lift. In both cases, the degree of the covering can be any number $d \geq 4$, provided that $d$ has the same parity of the Stiefel--Whitney number $w_1^4[X]$ in the case of $\RP^4$. Moreover, the branch set can be assumed to be non-singular if $d \geq 5$ and to have just nodal singularities if $d=4$.

math.GT↗
arXiv2025

A cut-and-paste mechanism to introduce fundamental group and construct new four-manifolds

We introduce a simple cut-and-paste mechanism to construct both orientable and nonorientable four-manifolds from a given initial one. This mechanism alters the fundamental group while preserving other essential topological invariants. It avoids codimension two cut-and-paste fundamental group computations and fast tracks the search for fixed-point free involutions. The mechanism proves useful to unveil novel exotic irreducible smooth structures on closed four-manifolds with finite cyclic fundamental group, which include $\Q$-homology real projective four-spaces.

math.GT↗
arXiv2025

Smooth structures on non-orientable $4$-manifolds via twisting operations

Four observations compose the main results of this note. The first records the existence of a smoothly embedded 2-sphere $S$ inside $\mathbb{R} P^2\times S^2$ such that performing a Gluck twist on $S$ produces a manifold $Y$ that is homeomorphic but not diffeomorphic to the total space of the non-trivial 2-sphere bundle over the real projective plane $S(2γ\oplus \mathbb{R})$. The second observation is that there is a 5-dimensional cobordism with a single 2-handle between the 4-manifold $Y$ and a mapping torus that was used by Cappell-Shaneson to construct an exotic $\mathbb{R} P^4$. This construction of $Y$ is similar to the one of the Cappell-Shaneson homotopy 4-spheres. The third observation is that twisting an embedded real projective plane inside $Y$ produces a manifold that is homeomorphic but not diffeomorphic to the circle sum of two copies of $\mathbb{R}P^4$. Knotting phenomena of 2-spheres in non-orientable 4-manifolds that stands in glaring contrast with known phenomena in the orientable domain is pointed out in the fourth observation.

math.GT↗
arXiv2025

$\text{Pin}^{\pm}$-structures on non-oriented 4-manifolds via Lefschetz fibrations

We study necessary and sufficient conditions for a 4-dimensional Lefschetz fibration over the 2-disk to admit a $\text{Pin}^{\pm}$-structure, extending the work of A. Stipsicz in the orientable setting. As a corollary, we get existence results of $\text{Pin}^{+}$ and $\text{Pin}^-$-structures on closed non-orientable 4-manifolds and on Lefschetz fibrations over the 2-sphere. In particular, we show via three explicit examples how to read-off $\text{Pin}^{\pm}$-structures from the Kirby diagram of a 4-manifold. We also provide a proof of the well-known fact that any closed 3-manifold $M$ admits a $\text{Pin}^-$-structure and we find a criterion to check whether or not it admits a $\text{Pin}^+$-structure in terms of a handlebody decomposition. We conclude the paper with a characterization of $\text{Pin}^+$-structures on vector bundles.

math.GT↗
arXiv2024

On Dold-Whitney's parallelizability of 4-manifolds

We present a proof of the fact that a closed orientable 4-manifold is parallelizable if and only if its second Stiefel-Whitney class, first Pontryagin class and Euler characteristics vanish. This follows from a stronger result due to Dold and Whitney on the classification of oriented sphere bundles over a 4-complex. The contribution of this note is to outline in detail an argument which is essentially due to R. Kirby, using the classification of $SO(4)$-bundles over the 4-sphere by means of their Euler and first Pontryagin classes as a main tool.

math.GT↗
arXiv2024

Some examples of small irreducible exotic 4-manifolds with free abelian fundamental group

We produce examples of pairwise non-diffeomorphic closed irreducible 4-manifolds with non-trivial free abelian fundamental group of rank less than three and small Euler characteristic. These exotic smooth structures become standard after taking a connected sum with a single copy of $S^2\times S^2$. The contributions of this paper include an explicit mechanism to computate the equivariant intersection form of 4-manifolds that are obtained via torus surgeries and a new stabilization result concerning exotic smooth structures with arbitrary fundamental group.

math.GT↗
arXiv2023

A recipe for exotic 2-links in closed 4-manifolds whose components are topological unknots

We describe a construction procedure of infinite sets of $2$-links in closed simply connected 4-manifolds that are topologically isotopic, smoothly inequivalent and componentwise topologically unknotted. These 2-links are the first examples of such kind in the literature. The examples provided have surface and free groups as their 2-link groups. We also point out an exotic Brunnian behaviour of such families, which highlights the important role of linking in creating exotic phenomena.

math.GT↗
arXiv2023

Existence results of Spin$(2,n-1)_0$-pseudo-Riemannian cobordisms

In this note, we study necessary and sufficient conditions for the existence of a Spin $(n + 1)$-dimensional cobordism that supports a non-singular and non-degenerate pseudo-Riemannian metric of signature $(2, n - 1)$, which restricts to a non-singular time-orientable Lorentzian metric on its boundary. The corresponding cobordism groups are computed.

math.GT↗