In this paper we define, for each aspherical orientable 3-manifold M endowed with a torus splitting T , a 2-dimensional fundamental l1-class [M ] whose l1norm has similar properties as the Gromov simplicial volume of M (additivity under torus splittings and isometry under finite covering maps). Next, we use the Gromov simplicial volume of M and the l1-norm of [M ] to give a complete characteriz...

If a GQ S ′ of order (s, s) is contained in a GQ S of order (s, s2) as a subquadrangle, then for each point X of S\S ′ the set of points OX of S ′ collinear with X form an ovoid of S ′. Thas and Payne proved that if S ′ = Q(4, q), q even, and OX is an elliptic quadric for each X ∈ S\S ′, then S ∼= Q(5, q). In this paper we provide a single proof for the q odd and q even cases by establishing a ...

Supergroupoids, double structures, and equivariant cohomology by Rajan Amit Mehta Doctor of Philosophy in Mathematics University of California, Berkeley Professor Alan Weinstein, Chair Q-groupoids and Q-algebroids are, respectively, supergroupoids and superalgebroids that are equipped with compatible homological vector fields. These new objects are closely related to the double structures of Ma...

Topological persistence methods provide a robust framework for analyzing large point cloud datasets topologically, and have been applied with great success towards homology computations on simplicial complexes. In this paper, we apply the persistence algorithm towards calculating a set of invariants related to the cup product structure on the cohomology ring for a space. These invariants expres...

certain algebraic invariants of the integral group ring zg of a group g were introduced and investigated in relation to the problem of extending the farrell-tate cohomology, which is defined for the class of groups of finite virtual cohomological dimension. it turns out that the finiteness of these invariants of a group g implies the existence of a generalized farrell-tate cohomology for g whic...

Bitmap images of arbitrary dimension may be formally perceived as unions of m-dimensional boxes aligned with respect to a rectangular grid in R. Cohomology and homology groups are well known topological invariants of such sets. Cohomological operations, such as the cup product, provide higher-order algebraic topological invariants, especially important for digital images of dimension higher tha...

We prove that for a topological operad P the operad of oriented cubical chains, C ∗ (P ), and the operad of singular chains, S∗(P ), are weakly equivalent. As a consequence, C ord ∗ (P ;Q) is formal if and only if S∗(P ;Q) is formal, thus linking together some formality results that are spread out in the literature. The proof is based on an acyclic models theorem for monoidal functors. We give ...

Our objective in this article is to show a possibly interesting structure of homotopic nature appearing in persistent (co)homology. Assuming that the filtration of the (say) simplicial set embedded in Rn induces a multiplicative filtration (which would not be a so harsh hypothesis in our setting) on the dg algebra given by the complex of simplicial cochains, we may use a result by T. Kadeishvil...