A homology stratification is a filtered space with local homology groups constant on strata. Despite being used by Goresky and MacPherson [3] in their proof of topological invariance of intersection homology, homology stratifications do not appear to have been studied in any detail and their properties remain obscure. Here we use them to present a simplified version of the Goresky–MacPherson pr...

We compute the equivariant $KO$-homology of classifying space for proper actions $\textrm{SL}_3(\mathbb{Z})$ and $\textrm{GL}_3(\mathbb{Z})$. also Bredon homology $K$-homology spaces $\textrm{PSL}_2(\mathbb{Z}[\frac{1}{p}])$ $\textrm{SL}_2(\mathbb{Z}[\frac{1}{p}])$ each prime $p$. Finally, we prove Unstable Gromov-Lawson-Rosenberg Conjecture a large class groups whose maximal finite subgroups a...

Abstract We introduce a new metric homology theory, which we call Moderately Discontinuous Homology, designed to capture Lipschitz properties of singular subanalytic germs. The main novelty our approach is allow “moderately discontinuous” chains, are specially advantageous for capturing the subtleties outer phenomena. Our invariant finitely generated graded abelian group any and homomorphisms ....

Recently, Ozsváth and Szabó introduced some algebraic constructions computing knot Floer homology in the spirit of bordered homology, including a family algebras $\mathcal B (n)$ and, for generator braid group on $n$ strands, certain t

This paper focuses on developing an efficient algorithm for analyzing a directed network (graph) from topological viewpoint. A prevalent technique such analysis involves computation of homology groups and their persistence. These concepts are well suited spaces that not directed. As result, one needs concept accommodates orientations in input space. Path-homology developed graphs by Grigoryan e...

We construct a stable homotopy refinement of quantum annular homology, link homology theory introduced by Beliakova, Putyra and Wehrli. For each $r\geq 2$ we associate to an $L$ naive $\mathbb{Z}/r\mathbb{Z}$-equivariant spectrum whose cohomology is isomorphic the as modules over $\mathbb{Z}[\mathbb{Z}/r\mathbb{Z}]$. The construction relies on equivariant version Burnside category approach Laws...

We give a new proof of the non-triviality wheel graph homology classes using higher operations on Lie and derived version Koszul duality for modular operads.