Multiplicative structure in equivariant cohomology

We introduce the notion of a strongly homotopy-comultiplicative resolution of a module coalgebra over a chain Hopf algebra, which we apply to proving a comultiplicative enrichment of a well-known theorem of Moore concerning the homology of quotient spaces of group actions. The importance of our enriched version of Moore's theorem lies in its application to the construction of useful cochain algebra models for computing multiplicative structure in equivariant cohomology. In the special cases of homotopy orbits of circle actions on spaces and of group actions on simplicial sets, we obtain small, explicit cochain algebra models that we describe in detail.Comment: 28 pages. Final version (cosmetic changes, slight reorganization), to appear in JPA

Homological thickness and stability of torus knots

In this paper we show that the non-alternating torus knots are homologically thick, i.e. that their Khovanov homology occupies at least three diagonals. Furthermore, we show that we can reduce the number of full twists of the torus knot without changing certain part of its homology, and consequently, we show that there exists stable homology of torus knots conjectured by Dunfield, Gukov and Rasmussen in \cite{dgr}. Since our main tool is the long exact sequence in homology, we have applied our approach in the case of the Khovanov-Rozansky ($sl(n)$) homology, and thus obtained analogous stability properties of $sl(n)$ homology of torus knots, also conjectured in \cite{dgr}.Comment: 24 pages, expanded Section

Improved bounds for the rate loss of multiresolution source codes

We present new bounds for the rate loss of multiresolution source codes (MRSCs). Considering an M-resolution code, the rate loss at the ith resolution with distortion D/sub i/ is defined as L/sub i/=R/sub i/-R(D/sub i/), where R/sub i/ is the rate achievable by the MRSC at stage i. This rate loss describes the performance degradation of the MRSC compared to the best single-resolution code with the same distortion. For two-resolution source codes, there are three scenarios of particular interest: (i) when both resolutions are equally important; (ii) when the rate loss at the first resolution is 0 (L/sub 1/=0); (iii) when the rate loss at the second resolution is 0 (L/sub 2/=0). The work of Lastras and Berger (see ibid., vol.47, p.918-26, Mar. 2001) gives constant upper bounds for the rate loss of an arbitrary memoryless source in scenarios (i) and (ii) and an asymptotic bound for scenario (iii) as D/sub 2/ approaches 0. We focus on the squared error distortion measure and (a) prove that for scenario (iii) L/sub 1/<1.1610 for all D/sub 2/<0.7250; (c) tighten the Lastras-Berger bound for scenario (i) from L/sub i//spl les/1/2 to L/sub i/<0.3802, i/spl isin/{1,2}; and (d) generalize the bounds for scenarios (ii) and (iii) to M-resolution codes with M/spl ges/2. We also present upper bounds for the rate losses of additive MRSCs (AMRSCs). An AMRSC is a special MRSC where each resolution describes an incremental reproduction and the kth-resolution reconstruction equals the sum of the first k incremental reproductions. We obtain two bounds on the rate loss of AMRSCs: one primarily good for low-rate coding and another which depends on the source entropy

The ghost length and duality on the chain and cochain type levels

We establish equalities between cochain and chain type levels of maps by making use of exact functors which connect appropriate derived and coderived categories. Relevant conditions for levels of maps to be finite are extracted from the equalities which we call duality on the levels. Moreover, we give a lower bound of the cochain type level of the diagonal map on the classifying space of a Lie group by considering the ghostness of a shriek map which appears in derived string topology. A variant of Koszul duality for a differential graded algebra is also discussed.Comment: 23 pages. This is a new verision of the preprint "Duality on the (co)chain type levels". The title is change

Relative measure homology and continuous bounded cohomology of topological pairs

Measure homology was introduced by Thurston in his notes about the geometry and topology of 3-manifolds, where it was exploited in the computation of the simplicial volume of hyperbolic manifolds. Zastrow and Hansen independently proved that there exists a canonical isomorphism between measure homology and singular homology (on the category of CW-complexes), and it was then shown by Loeh that, in the absolute case, such isomorphism is in fact an isometry with respect to the L^1-seminorm on singular homology and the total variation seminorm on measure homology. Loeh's result plays a fundamental role in the use of measure homology as a tool for computing the simplicial volume of Riemannian manifolds. This paper deals with an extension of Loeh's result to the relative case. We prove that relative singular homology and relative measure homology are isometrically isomorphic for a wide class of topological pairs. Our results can be applied for instance in computing the simplicial volume of Riemannian manifolds with boundary. Our arguments are based on new results about continuous (bounded) cohomology of topological pairs, which are probably of independent interest.Comment: 35 page