In this paper certain filtrations of topological Hochschild homology and topological cyclic homology are examined. As an example we show how the filtration with respect to a nilpotent ideal gives rise to an analog of a theorem of Goodwillie saying that rationally relative K -theory and relative cyclic homology agree. Our variation says that the p-torsion parts agree in a range of degrees. We us...

We show that the π-equivariant chain complex (π = π1(M(A))), C•(X̃), associated to a Morse-theoretic minimal CW -structure X on the complement M(A) of an arrangement A, is independent of X . The same holds for all scalar extensions, C•(X̃)⊗Zπ KZ, K a field, where X is an arbitrary minimal CW -structure on a space M . When A is a section of another arrangement Â, we show that the divisibility prop...

We show that a spectral sequence developed by Lipshitz and Treumann, for application to Heegaard Floer theory, converges localized form of topological Hochschild homology with coefficients. This allows us the target this can be identified when is torsion-free as module over $\mathrm{THH}_*(\mathbb{F}_2)$, parallel results Mathew on degeneration Hodge-to-de Rham sequence. To carry out, we apply ...

Abstract The combination of persistent homology and discrete Morse theory has proven very effective in visualizing analyzing big heterogeneous data. Indeed, topology provides computable coarse summaries data independently from specific coordinate systems does so robustly to noise. Moreover, the geometric content a gradient vector field is useful for visualization purposes. case multivariate sti...

Heegaard Floer homology, first introduced by P. Ozsváth and Z. Szabó in [OS04b], associates to a 3-manifold Y a family of relatively graded abelian groups HF (Y, t), indexed by Spin structures t on Y . In the case that Y is a rational homology sphere, Ozsváth and Szabó lift the relative Z-grading to an absolute Q-grading [OS06]. This induces a relative Q-grading on ⊕ t∈Spin(Y ) HF (Y, t). In th...

Abstract We construct secondary cup and cap products on coarse (co-)homology theories from given cross slant products. They are defined for spaces relative to weak generalized controlled deformation retracts. On ordinary cohomology, our product agrees with a by Roe. For coarsifications of topological theories, correspond the primary Higson dominated coronas via transgression maps. And in case $...

In this article, we show the existence of certain exact sequences with respect to two homology theories, called d-homology and extended d-homology. We present sufficient conditions for the existence of long exact extended d- homology sequence. Also we give some illustrative examples.

We develop a method for measuring and localizing homology classes. This involves two problems. First, we define relevant notions of size for both a homology class and a homology group basis, using ideas from relative homology. Second, we propose an algorithm to compute the optimal homology basis, using techniques from persistent homology and finite field algebra. Classes of the computed optimal...

We develop a method for measuring and localizing homology classes. This involves two problems. First, we define relevant notions of size for both a homology class and a homology group basis, using ideas from relative homology. Second, we propose an algorithm to compute the optimal homology basis, using techniques from persistent homology and finite field algebra. Classes of the computed optimal...