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We establish a 3-manifold invariant for each finite-dimensional, involutory Hopf algebra. If the Hopf algebra is the group algebra of a group G, the invariant counts homomorphisms from the fundamental group of the manifold to G. The invariant can be viewed as a state model on a Heegaard diagram or a triangulation of the manifold. The computation of the invariant involves tensor products and contractions of the structure tensors of the algebra. We show that every formal expression involving these tensors corresponds to a unique 3-manifold modulo a well-understood equivalence. This raises the possibility of an algorithm which can determine whether two given 3-manifolds are homeomorphic. 1

Persistent homology is an algebraic tool for measuring topological features of shapes and functions. It casts the multi-scale organization we frequently observe in nature into a mathematical formalism. Here we give a record of the short history of persistent homology and present its basic concepts. Besides the mathematics we focus on algorithms and mention the various connections to applications, including to biomolecules, biological networks, data analysis, and geometric modeling.

Abstract: We develop a calculus of surgery data, called bridged links, which involves besides links also pairs of balls that describe one-handle attachements. As opposed to the usual link calculi of Kirby and others this description uses only elementary, local moves(namely modifications and isolated cancellations), and it is valid also on non-simply connected and disconnected manifolds. In particular, it allows us to give a presentation of a 3-manifold by doing surgery on any other 3-manifold with the same boundary. Bridged link presentations on unions of handlebodies are used to give a Cerf-theoretical derivation of presentations of 2+1-dimensional cobordisms categories in terms of planar ribbon tangles and their composition rules. As an application we give a different, more natural proof of the Matveev-Polyak presentations of the mapping class group, and, furthermore, find systematically surgery presentations of general mapping tori. We discuss a natural extension of the Reshetikhin Turaev invariant to the calculus of bridged links. Invariance follows now- similar as for knot invariants- from simple identifications of the elementary moves with elementary categorial relations for invariances or cointegrals, respectively. Hence, we avoid the lengthy computations and the unnatural Fenn-Rourke reduction of the original

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We prove that the 2-groupoid of transformations of rigid structures on surfaces has a finite presentation, establishing a result first conjectured by Moore and Seiberg. We also show that a finite dimensional, unitary, cyclic topological quantum field theory gives rise to a representation of this 2-groupoid.

Abstract. The decomposition of an arbitrary axiomatic topological quantum field theory or TQFT into indecomposable theories is given. In particular, unitary TQFT’s in arbitrary dimensions are shown to decompose into a sum of theories in which the Hilbert space of the sphere is one-dimensional, and indecomposable two-dimensional theories are classified. 1.

Introduction. (0.1) The Stasheff polytope, or associahedron, Kn, is a convex polytope of dimension n − 2 whose vertices correspond to complete parenthesizings of the product of n factors x1,..., xn. It was introduced by J. Stasheff [St] in his study of homotopy associativity for binary multiplications on topological spaces. In this paper we describe a surprising appearance of Stasheff

Abstract. We construct functor-valued invariants of three-dimensional cobordisms using Lagrangian Floer theory, mapping the Donaldson-Fukaya (or Fukaya) category of moduli spaces of bundles with degree coprime to the rank on the incoming boundary to that for the outgoing boundary. These are Lagrangian Floer versions of instanton Floer homology of three manifolds in the reducible-free case. Contents

Abstract. We provide explicit, simple, geometric formulas for free involutions ρ of Euclidean spheres that are not conjugate to the antipodal involution. Therefore the quotient S n /ρ is a manifold that is homotopically equivalent but not diffeomorphic to RP n. We use these formulas for constructing explicit non-trivial elements in π1Diff(S 5) and π1Diff(S 13) and to provide explicit formulas for non-cancellation phenomena in group actions. 1.