On some standard constructions in algebraic topology

Some Algebraic Definitions and Constructions

Definition 1. A monoid is a set M with an element e and an associative multiplication M×M −→ M for which e is a two-sided identity element: em = m = me for all m ∈ M . A group is a monoid in which each element m has an inverse element m, so that mm = e = mm. A homomorphism f : M −→ N of monoids is a function f such that f(mn) = f(m)f(n) and f(eM ) = eN . A “homomorphism” of any kind of algebrai...

Some New Algebraic Methods in Topology

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...

Lecture 5: Some Basic Constructions in Symplectic Topology

In complex geometry the blowing-up operation amounts to replace a point in a space by the space of complex tangent lines through that point. It is a local operation which can be explicitly written down as follows. Consider the blow-up of Cn at the origin. This is the complex submanifold of Cn × CPn−1 C̃ ≡ {((z1, z2, · · · , zn), [w1, w2, · · · , wn])|(z1, z2, · · · , zn) ∈ [w1, w2, · · · , wn]}....

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...