International Journal of Mathematics

We study the topology of line singularities, which are complex hypersurface germs with non-isolated singularity given by a smooth curve. We describe the degeneration of its Milnor fiber to the singular hypersurface by means of a vanishing polyhedron in the Milnor fiber. As a milestone, we also study the topology of the degeneration of a complex isolated singularity hypersurface under a nonlocal point of view.

In this paper, we examine the analytic bilateral Laplace–Feynman transform (BLFT) for functions on the Hilbert space H. We then proceed to establish a relationship between the analytic BLFT on H and the analytic Fourier–Feynman transform on the abstract Wiener space B.

In this paper, we discuss a relationship between the surface symmetry and the spectral asymmetry. More precisely we show that an automorphism of the Macbeath surface of genus 7, or one of the three Hurwitz surfaces of genus 14 is reducible if and only if the η-invariant of the corresponding mapping torus vanishes.

Given a group G and a class of manifolds 𝒞 (e.g. symplectic, contact, Kähler, etc.), it is an old problem to find a manifold M_G ∈ 𝒞 whose fundamental group is G. This article refines it: for a group G and a positive integer r find M_G ∈ 𝒞 such that π₁(M_G) = G and π_i(M_G) = 0 for 1 < i < r. We thus provide a unified point of view systematizing known and new results in this direction for various different classes of manifolds. The largest r for which such an M_G ∈ 𝒞 can be found is called the homotopical height ht_𝒞(G). Homotopical height provides a dimensional obstruction to finding a K(G, 1) space within the given class 𝒞, leading to a hierarchy of these classes in terms of "softness" or "hardness" à la Gromov. We show that the classes of closed contact, CR, and almost complex manifolds as well as the class of (open) Stein manifolds are soft. The classes 𝒮𝒫 and 𝒞𝒜 of closed symplectic and complex manifolds exhibit intermediate "softness" in the sense that every finitely presented group G can be realized as the fundamental group of a manifold in 𝒮𝒫 and a manifold in 𝒞𝒜. For these classes, ht_𝒞(G) provides a numerical invariant for finitely presented groups. We give explicit computations of these invariants for some standard finitely presented groups. We use the notion of homotopical height within the "hard" category of Kähler groups to obtain partial answers to questions of Toledo regarding second cohomology and second group cohomology of Kähler groups. We also modify and generalize a construction due to Dimca, Papadima and Suciu to give a potentially large class of projective groups violating property FP.