# Strong Homology Theory of Continuous Maps

## TL;DR

This paper develops a strong homology theory for continuous maps, establishing it as a shape invariant with semi-continuity, based on prior spectral and homology group definitions.

## Contribution

It introduces a new strong homology functor for continuous maps, proving its invariance and semi-continuity, and formulates axioms for its uniqueness.

## Key findings

- Strong homology groups form a homology type functor.
- The functor is a strong shape invariant.
- It possesses the semi-continuous property.

## Abstract

The current work is motivated by the papers $[B_3]$, $[B_6]$, $[Be]$, $[Be-Tu]$. In particular, using Theorem 3.7 of $[B_3]$ and methods developed in this paper, the spectral and strong homology groups of continuous maps were defined and studied $[B_6]$, $[Be]$, $[Be-Tu]$. In this paper we will show that strong homology groups of continuous maps are a homology type functor, which is a strong shape invariant and has the semi-continuous property. We will formulate the new axioms and the conjunction on the uniqueness of the constructed functor.