Let X be a simply connected space and LX the space of free loops on X. We determine the mod p cohomology algebra of LX when the mod p cohomology of X is generated by one element or is an exterior algebra on two generators. We also provide lower bounds on the dimensions of the Hodge decomposition factors of the rational cohomology of LX when the rational cohomology of X is a graded complete intersection algebra. The key to both of these results is the identification of an important subalgebra of...

Using the path lattice cohomology we provide a conceptual topological characterization of the geometric genus for certain complex normal surface singularities with rational homology sphere links, which is uniformly valid for all superisolated and Newton non-degenerate hypersurface singularities.

Let M be a closed orientable manifold of dimension dand ${𝒞}^{*}\left(M\right)$ be the usual cochain algebra on M with coefficients in a fieldk. The Hochschild cohomology of M, $H{H}^{*}({𝒞}^{*}\left(M\right);{𝒞}^{*}\left(M\right))$ is a graded commutative and associative algebra. The augmentation map $\epsilon :{𝒞}^{*}\left(M\right)\to 𝑘$ induces a morphism of algebras $I:H{H}^{*}({𝒞}^{*}\left(M\right);{𝒞}^{*}\left(M\right))\to H{H}^{*}({𝒞}^{*}\left(M\right);𝑘)$. In this paper we produce a chain model for the morphism I. We show that the kernel of I is a nilpotent ideal and that the image of I is contained in the center of $H{H}^{*}({𝒞}^{*}\left(M\right);𝑘)$, which is in general quite small. The algebra $H{H}^{*}({𝒞}^{*}\left(M\right);{𝒞}^{*}\left(M\right))$ is expected to be isomorphic...

In this paper we extend the Eilenberg-Steenrod axiomatic description of a homology theory from the category of topological spaces to an arbitrary category and, in particular, to a topos. Implicit in this extension is an extension of the notions of homotopy and excision. A general discussion of such homotopy and excision structures on a category is given along with several examples including the interval based homotopies and, for toposes, the excisions represented by “cutting out” subobjects. The...

Khovanov homology for knots has generated a flurry of activity in the topology community. This paper studies the Khovanov type cohomology for graphs with a special attention to torsion. When the underlying algebra is ℤ[x]/(x²), we determine precisely those graphs whose cohomology contains torsion. For a large class of algebras, we show that torsion often occurs. Our investigation of torsion led to other related general results. Our computation could potentially be used to predict the Khovanov-Rozansky...

The one-term distributive homology was introduced in [Prz] as an atomic replacement of rack and quandle homology, which was first introduced and developed by Fenn-Rourke-Sanderson [FRS] and Carter-Kamada-Saito [CKS]. This homology was initially suspected to be torsion-free [Prz], but we show in this paper that the one-term homology of a finite spindle may have torsion. We carefully analyze spindles of block decomposition of type (n,1) and introduce various techniques to compute their homology precisely....