Some new algebraic methods in topology

Some New Algebraic Methods in Topology

Quantum Methods in Algebraic Topology

In this paper, we present a new version of cochains in Algebraic Topology, starting with “quantum differential forms”. This version provides many examples of modules over the braid group, together with a control of the non commutativity of cup-products on the cochain level. If the quantum parameter q is equal to 1, we essentially recover the commutative differential graded algebra of de Rham-Su...

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 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...

Categorically-algebraic topology and its applications

This paper introduces a new approach to topology, based on category theory and universal algebra, and called categorically-algebraic (catalg) topology. It incorporates the most important settings of lattice-valued topology, including poslat topology of S.~E.~Rodabaugh, $(L,M)$-fuzzy topology of T.~Kubiak and A.~v{S}ostak, and $M$-fuzzy topology on $L$-fuzzy sets of C.~Guido. Moreover, its respe...