We define a notion of cosheaves on diffeological spaces by cosheaves on the site of plots. This provides a framework to describe diffeological objects such as internal tangent bundles, the Poincar'{e} groupoids, and furthermore, homology theories such as cubic homology in diffeology by the language of cosheaves. We show that every cosheaf on a diffeological space induces a cosheaf in terms of t...

Quasi-alternating links are a natural generalization of alternating links. In this paper, we show that quasi-alternating links are “homologically thin” for both Khovanov homology and knot Floer homology. In particular, their bigraded homology groups are determined by the signature of the link, together with the Euler characteristic of the respective homology (i.e. the Jones or the Alexander pol...

We present computational results about quasi-alternating knots and links and odd homology obtained by looking at link families in the Conway notation. More precisely, we list quasi-alternating links up to 12 crossings and the first examples of quasi-alternating knots and links with at least two different minimal diagrams, where one is quasi-alternating and the other is not. We provide examples ...

We prove that the length of any gap in differential grading Khovanov homology quasi-alternating link is one. As a consequence, we obtain Jones polynomial such This establishes weaker version Conjecture 2.3 (Topol Appl 264:1–11, 2019). Moreover, lower bound for determinant terms breadth its polynomial. 3.8 (Algebr Geom Topol 15:1847–1862, 2015). The main tool obtaining this result establishing K...

The Exel-Loring formula asserts that two topological invariants associated to a pair of almost commuting unitary matrices coincide. Such can be viewed as quasi-representation $${\mathbb {Z}}^2$$ . We give generalization this for countable discrete groups. also show the nontriviality corresponding quasidiagonal groups which are coarsely embeddable in Hilbert space and have nonvanishing second Be...

We develop the theory of parity quasi-complexes (PQC), preparing the set up for defining derived functors using resolutions in the nonabelian case. A homotopy structure on the category of PQC is defined, yielding a 2-category structure. The nonabelian homology functor factors through the corresponding homotopy category. Following the relative homological algebra approach, resolutions are define...

In their recent proof [2] of Schapira-Schneider’s conjecture [19], Bressler-Nest-Tsygan construct a (generalized) Chern character from the Ktheory of perfect complexes to the negative cyclic homology HC− ∗ (A) of a sheaf of algebras A on a topological space. The first aim of this paper is to show how to construct a (classical) Chern character defined on the Grothendieck group of perfect complex...

This paper constitutes a preliminary discussion of joint work in progress as presented by the first author at Stony Brook in June 1991 at the symposium in honor of John Milnor. Our results include a construction of an intersection pairing on spaces of algebraic cycles on a given smooth complex quasi-projective variety, thereby providing a ring structure in “Lawson homology.” We verify that the ...