A-homology, A-homotopy and spectral sequences

TL;DR

This paper introduces an A-shaped homology theory and a spectral sequence for A-homotopy groups, generalizing classical results and providing new computational tools in algebraic topology.

Contribution

It develops a new A-homology theory and a relative Federer spectral sequence, extending classical theorems to A-homotopy contexts.

Findings

A new A-homology theory with favorable properties.

A relative Federer spectral sequence for A-homotopy groups.

Generalization of the Hopf-Whitney theorem.

Abstract

Given a CW-complex A we define an `A-shaped' homology theory which behaves nicely towards A-homotopy groups allowing the generalization of many classical results. We also develop a relative version of the Federer spectral sequence for computing A-homotopy groups. As an application we derive a generalization of the Hopf-Whitney theorem.