Straight homotopy invariants

TL;DR

This paper introduces the concept of straight homotopy invariants, showing they can be universally expressed through a homological invariant, thus linking homotopy and homology in a new way.

Contribution

It defines straight homotopy invariants and proves they can be represented by a universal homological invariant, advancing the understanding of homotopy invariants.

Findings

All straight invariants are expressible via a universal homological invariant.

The paper establishes a connection between homotopy invariants and homology.

It characterizes the structure of admissible homomorphisms in homotopy theory.

Abstract

Let $X$ and $Y$ be spaces and $M$ be an abelian group. A homotopy invariant $f\colon [X,Y]\to M$ is called straight if there exists a homomorphism $F\colon L(X,Y)\to M$ such that $f([a])=F(\langle a\rangle)$ for all $a\in C(X,Y)$. Here $\langle a\rangle\colon\langle X\rangle\to\langle Y\rangle$ is the homomorphism induced by $a$ between the abelian groups freely generated by $X$ and $Y$ and $L(X,Y)$ is a certain group of `admissible' homomorphisms. We show that all straight invariants can be expressed through a `universal' straight invariant of homological nature.