# The homotopy invariance of dihedral homology of involutive $A_\infty$-algebras over rings

## TL;DR

This paper constructs a dihedral homology functor for involutive $A_
abla$-algebras over rings and proves it is invariant under homotopy equivalences, extending the understanding of algebraic invariants in homotopical algebra.

## Contribution

It introduces a dihedral homology functor for involutive $A_
abla$-algebras and proves its homotopy invariance over rings, a novel extension in the field.

## Key findings

- Constructed the dihedral homology functor for involutive $A_
abla$-algebras.
- Proved the functor preserves homotopy equivalences as isomorphisms.
- Extended the invariance properties of dihedral homology to a broader algebraic context.

## Abstract

The dihedral homology functor $HD:A_\infty^{{\rm inv}}(K)\to GrM(K)$ from the category $A_\infty^{{\rm inv}}(K)$ of involutive $A_\infty$-algebras over any commutative unital ring $K$ to the category $GrM(K)$ of graded $K$-modules is constructed. Further, it is showed that this functor sends homotopy equivalences of involutive $A_\infty$-algebras into isomorphisms of graded modules.