The analytic index of the operator d+δ on a compact orientable Riemannian manifold M is the Euler characteristic of M, χ (M) . This paper extends this result to a class of complete noncompact manifolds, those with finitely generated rational homology and finitely many quasi-periodic ends. The latter term means that there is a neighborhood of each end which is quasi-isometric to a neighborhood o...

For any 1-reduced simplicial set K we define a canonical, coassociative coproduct on ΩC(K), the cobar construction applied to the normalized, integral chains on K, such that any canonical quasi-isomorphism of chain algebras from ΩC(K) to the normalized, integral chains on GK, the loop group of K, is a coalgebra map up to strong homotopy. Our proof relies on the operadic description of the categ...

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.

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 prove a sheaf-theoretic derived-category generalization of GreenleesMay duality (a far-reaching generalization of Grothendieck’s local duality theorem): for a quasi-compact separated scheme X and a “proregular” subscheme Z—for example, any separated noetherian scheme and any closed subscheme—there is a sort of sheafified adjointness between local cohomology supported in Z and left-derived co...

Building upon work of Y. Shalom we give a homological-algebra flavored definition of an induction map in group homology associated to a topological coupling. As an application we obtain that the cohomological dimension cdR over a commutative ring R satisfies the inequality cdR(Λ) ≤ cdR(Γ) if Λ embeds uniformly into Γ and cdR(Λ) < ∞ holds. Another consequence of our results is that the Hirsch ra...

Q-Gorenstein toric contact manifolds provide an interesting class of examples with torsion first Chern class. They are completely determined by certain rational convex polytopes, called diagrams, and arise both as links isolated singularities prequantizations monotone symplectic orbifolds. In this paper we show how the cylindrical homology invariants a manifold related to Ehrhart (quasi-)polyno...

We investigate the large λ behavior of σ((p−λA)2) when the zero set of B = dA has a non-empty interior. With certain technical hypotheses we show that if either B is bounded away from zero for large |x| or periodic and certain quotients of standard homology groups are finite rank, then σ((p − λA)2) approaches a quasi-periodic orbit in the space of subsets of [0,∞).

In the previous lecture, we outlined some approaches to describing the cohomology of the classifying space of G-bundles M on a Riemann surface X. For example, we asserted that the cochain complex C * (M; Q) is quasi-isomorphic to a continuous tensor product ⊗ x∈X C * (BG x ; Q). Here it is vital that we work at the level of cochains, rather than cohomology: there is no corresponding procedure t...

We compare various groups of 0-cycles on quasi-projective varieties over a field. As applications, we show that for certain singular projective varieties, the Levine-Weibel Chow group coincides with corresponding Friedlander-Voevodsky motivic cohomology. also an algebraically closed field positive characteristic, modulus smooth variety respect to reduced divisor Suslin homology complement divis...