arXiv subjects
Publications and source records attributed to Rafael Torres.
We produce new collections of homotopy 4-spheres and homotopy real projective 4-spaces by extending a classical construction due to Cappell-Shaneson. We use the extension to devise a mechanism that produces sets of smooth 4-manifolds within a given homeomorphism class that satisfies a mild topological condition.
For each odd integer $p > 1$, we construct infinitely many pairwise non-diffeomorphic irreducible smooth structures on a definite 4-manifold with infinite fundamental group whose abelianization is $\Z/2p\Z\times \Z/2\Z$.
We show that the classification up to homeomorphism of closed topological nonorientable 4-manifolds with fundamental group of order 2 due to Hambleton-Kreck-Teichner can be used to classify a large set of such 4-manifolds with cyclic fundamental group of order 2p for every odd $p > 1$. This is done through a simple cut-and-paste construction, and classical and modified surgery theory are used only through results of Hambleton-Kreck-Teichner and Khan. It is plausible that this set comprises all closed topological nonorientable 4-manifolds with $\pi_1 = \Z/2p$. We collect several interesting questions whose answers would guarantee a complete classification.
We show that any finitely presented group with an index two subgroup is realized as the fundamental group of a closed smooth non-orientable four-manifold that admits an exotic smooth structure, which is obtained by performing a Gluck twist. The orientation 2-covers of these four-manifolds are diffeomorphic. These two smooth structures remain inequivalent after adding arbitrarily many copies of the product of a pair of 2-spheres and stabilize after adding a single copy of the complex projective plane.
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.
We point out that any stable generalized complex structure on a sphere bundle over a closed surface of genus at least two must be of constant type.
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.
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 $S(2\gamma \oplus \mathbb{R})$. 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.
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.
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.
We describe a procedure to construct infinite sets of pairwise smoothly inequivalent 2-spheres in simply connected 4-manifolds, which are topologically isotopic and whose complement has a prescribed fundamental group that satisfies some conditions. This class of groups include finite cyclic groups and the binary icosahedral group. These are the first known examples of knotting phenomena in 4-manifolds with such properties. Examples of locally flat embedded 2-spheres in non-smoothable 4-manifolds are also given.
We discuss the geography problem of closed oriented 4-manifolds that admit a Riemannian metric of positive scalar curvature, and use it to survey mathematical work employed to address Gromov's observation that manifolds with positive scalar curvature tend to be inessential by focusing on the four-dimensional case. We also point out an strengthening of a result of Carr and its extension to the non-orientable realm.
Using recent work of Bettiol, we show that a first-order conformal deformation of Wilking's metric of almost-positive sectional curvature on $S^2\times S^3$ yields a family of metrics with strictly positive average of sectional curvatures of any pair of 2-planes that are separated by a minimal distance in the 2-Grassmanian. A result of Smale's allows us to conclude that every closed simply connected 5-manifold with torsion-free homology and trivial second Stiefel-Whitney class admits a Riemannian metric with a strictly positive average of sectional curvatures of any pair of orthogonal 2-planes.
We point out that recent constructions of inequivalent smooth structures yield a manufacturing procedure of infinite sets of pairwise smoothly non-isotopic nullhomologous 2-tori and spheres inside a myriad of 4-manifolds. The corresponding infinite set consists of topologically isotopic surfaces that topologically bound a handlebody in several instances.
We describe a construction of Riemannian metrics of nonnegative sectional curvature on a closed smooth nonorientable 4-manifold with fundamental group of order two that realizes a homotopy class that was not previously known to contain nonnegatively curved manifolds. The procedure yields new metrics of nonnegative sectional curvature on any 2-sphere bundle with base space the 2-sphere or the real projective plane.
Typically the use of the Rayleigh-Sommerfeld diffraction formula as a photon propagator is widely accepted due to the abundant experimental evidence that suggests that it works. However, a direct link between the propagation of the electromagnetic field in classical optics and the propagation of photons where the square of the probability amplitude describes the transverse probability of the photon detection is still an issue to be clarified. We develop a mathematical formulation for the photon propagation using the formalism of electromagnetic field quantization and the path-integral method, whose main feature is its similarity with a fractional Fourier transform (FrFT). Here we show that, because of the close relation existing between the FrFT and the Fresnel diffraction integral, this propagator can be written as a Fresnel diffraction, which brings forward a discussion of the fundamental character of it at the photon level compared to the Huygens-Fresnel principle. Finally, we carry out an experiment of photon counting by a rectangular slit supporting the result that the diffraction phenomenon in the Fresnel approximation behaves as the actual classical limit.
We prove that the existence of totally real immersions of manifolds is a closed property under cut-and-paste constructions along submanifolds including connected sums. We study the existence of totally real embeddings for simply connected 5-manifolds and orientable 6-manifolds and determine the diffeomorphism and homotopy types. We show that the fundamental group is not an obstruction for the existence of a totally real embedding for high-dimensional manifolds in contrast with the situation in dimension four.
We show that a pseudo-holomorphic embedding of an almost-complex $2n$-manifold into almost-complex $(2n + 2)$-Euclidean space exists if and only if there is a CR regular embedding of the $2n$-manifold into complex $(n + 1)$-space. We remark that the fundamental group does not place any restriction on the existence of either kind of embedding when $n$ is at least three. We give necessary and sufficient conditions in terms of characteristic classes for a closed almost-complex 6-manifold to admit a pseudo-holomorphic embedding into $\R^8$ equipped with an almost-complex structure that need not be integrable.