In this paper, we prove an $\mathbb{A}^1$-homology version of the Whitehead theorem with dimension bound. We also excision for $\mathbb{A}^1$-homology, Suslin homology, and $\mathbb{A}^1$-homotopy sheaves. order to these results, develop a general theory relative sheaves establish weak comparison result between $\mathbb{A}^1$- homology. As example as application results above, compute hyperplan...

In this paper we present a new approach to computing homology (with field coefficients) and persistent homology. We use concepts from discrete Morse theory, to provide an algorithm which can be expressed solely in terms of simple graph theoretical operations. We use iterated Morse decomposition, which allows us to sidetrack many problems related to the standard discrete Morse theory. In particu...

The purpose of this paper is to calculate the cyclic homology of rings of integers of global fields. We accomplish this by explicitly computing the homology of the simple complex associated to Tsygan’s double complex. To accomplish this, we first compute the cyclic homology of cyclic algebras, i.e., algebras of the form A = R[t]/(P (t)), where P is a monic polynomial with coefficients in R. Mor...

As a first step towards computing instanton-generated superpotentials in heterotic standard model vacua, we determine the Gromov-Witten invariants for a Calabi-Yau threefold with fundamental group π1(X) = Z3 × Z3. We find that the curves fall into homology classes in H2(X,Z) = Z⊕ (Z3 ⊕Z3). The unexpected appearance of the finite torsion subgroup in the homology group complicates our analysis. H...

We introduce a homology theory for k-graphs and explore its fundamental properties. We establish connections with algebraic topology by showing that the homology of a k-graph coincides with the homology of its topological realisation as described by Kaliszewski et al. We exhibit combinatorial versions of a number of standard topological constructions, and show that they are compatible, from a h...