On the topology of some even-dimensional algebraic hypersurfaces

The Topology of Complex Hypersurfaces

Let f be a polynomial in n+1 complex variables. In this paper, we study the topology of the set V = f−1(0). While V is a differentiable manifold in the neighborhood of simple points, the topology is more complicated at singular points. We use the following construction to study the topology at singular points. Let S be a small sphere around a singular point, and let K = S ∩V . We will prove tha...

Duality, Manifolds and Some Algebraic Topology

This is a short note intended to explore the applications of duality theory to the study of manifolds. I discuss Alexander duality, Lefschetz duality and Poincare duality, along with applications to the study of compact connected orientable manifolds. 1. Manifolds and points 1.1. The core question. One of the questions we shall be interested in is: Given two manifolds M and N , what are the way...

Some New Algebraic Methods in Topology

Some Congruences Motivated by Algebraic Topology

In July 2005, D. M. Davis posted several original numbertheoretic conjectures arising from his investigation of homotopy exponents of the special unitary group SU(n). The recent study of these conjectures and their generalizations by the author and Davis led to a strong lower bound for the homotopy p-exponent of SU(n), as well as a number of new combinatorial congruences some of which extend va...

Some Remarks on Scattering Resonances in Even Dimensional Euclidean Scattering

The purpose of this paper is to prove some results about quantum mechanical black box scattering in even dimensions d ≥ 2. We study the scattering matrix and prove some identities which hold for its meromorphic continuation onto Λ, the Riemann surface of the logarithm function. We study the multiplicities of the poles of the continued scattering matrix on each sheet of Λ and relate these to the...