On torsion of non-acyclic cellular chain complexes of even manifolds in a unique factorisation monoid
Esma Dirican Erdal
TL;DR
This work addresses the problem of multiplicativity of Reidemeister–Franz torsion for high-dimensional, non-acyclic manifolds arising from connected sums. It builds on Milnor’s multiplicativity framework and uses Mayer–Vietoris and symplectic chain-complex techniques to relate the torsion of punctured and unpunctured pieces, computing basic building blocks such as the closed unit ball and twofold connected sums. The main contribution is a no-correction multiplicativity result for the class $\mathcal{M}_{2n}^{\mathrm{Diff},\mathrm{hc}}$ in dimensions with $n\equiv 3,5,7\pmod{8}$ and $n\neq 15,31$, showing $\mathbb{T}_{RF}(W_p^{2n})$ factors as the product of the torsions of irreducible summands. This extends high-dimensional torsion multiplicativity beyond acyclic cases and illuminates how geometric decompositions translate into algebraic torsion invariants with potential implications for understanding manifold structure in high dimensions.
Abstract
Let $\mathcal{M}_{2n}^{\mathrm{Diff},\mathrm{hc}}$ be a multiplicative factorisation monoid over highly connected differentiable closed connected oriented manifolds. Any $2n$-dimensional manifold $W_p^{2n}$ from $\mathcal{M}_{2n}^{\mathrm{Diff},\mathrm{hc}}$ admits a unique connected sum decomposition into manifolds that cannot be decomposed any further. By using this decomposition, we prove that Reidemeister-Franz torsion of $W_p^{2n}$ can be written as the product of Reidemeister-Franz torsions of the manifolds in the decomposition without the corrective term.