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40
Higherdimensional algebra and topological quantum field theory
 Jour. Math. Phys
, 1995
"... For a copy with the handdrawn figures please email ..."
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Cited by 140 (14 self)
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For a copy with the handdrawn figures please email
Involutory Hopf algebras and 3manifold invariants
 Intern. J. Math
, 1991
"... We establish a 3manifold invariant for each finitedimensional, 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 ..."
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Cited by 39 (4 self)
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We establish a 3manifold invariant for each finitedimensional, 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 3manifold modulo a wellunderstood equivalence. This raises the possibility of an algorithm which can determine whether two given 3manifolds are homeomorphic. 1
Persistent Homology  a Survey
 CONTEMPORARY MATHEMATICS
"... Persistent homology is an algebraic tool for measuring topological features of shapes and functions. It casts the multiscale 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. ..."
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Cited by 36 (1 self)
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Persistent homology is an algebraic tool for measuring topological features of shapes and functions. It casts the multiscale 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.
Bridged links and tangle presentations of cobordism categories
 Adv. Math
, 1999
"... Abstract: We develop a calculus of surgery data, called bridged links, which involves besides links also pairs of balls that describe onehandle attachements. As opposed to the usual link calculi of Kirby and others this description uses only elementary, local moves(namely modifications and isolated ..."
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Cited by 23 (5 self)
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Abstract: We develop a calculus of surgery data, called bridged links, which involves besides links also pairs of balls that describe onehandle 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 nonsimply connected and disconnected manifolds. In particular, it allows us to give a presentation of a 3manifold by doing surgery on any other 3manifold with the same boundary. Bridged link presentations on unions of handlebodies are used to give a Cerftheoretical derivation of presentations of 2+1dimensional cobordisms categories in terms of planar ribbon tangles and their composition rules. As an application we give a different, more natural proof of the MatveevPolyak 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 FennRourke reduction of the original
On the groupoid of transformations of rigid structures on surfaces
 J. MATH. SCI. UNIV. TOKYO
, 1999
"... We prove that the 2groupoid 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 2g ..."
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Cited by 10 (6 self)
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We prove that the 2groupoid 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 2groupoid.
Direct sum decompositions and indecomposable TQFTs
 J. Math. Phys
, 1995
"... 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 onedimensional, and i ..."
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Cited by 9 (0 self)
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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 onedimensional, and indecomposable twodimensional theories are classified. 1.
Concordance spaces, higher simple homotopy theory, and applications
 Proc. Sympos. Pure
, 1978
"... While much is now known, through surgery theory, about the classification problem for manifolds of dimension at least five, information about the automorphism groups of such manifolds is as yet rather sparse. In fact, it seems that there is not a single closed manifold M of dimension greater than th ..."
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Cited by 8 (0 self)
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While much is now known, through surgery theory, about the classification problem for manifolds of dimension at least five, information about the automorphism groups of such manifolds is as yet rather sparse. In fact, it seems that there is not a single closed manifold M of dimension greater than three for which the homotopy type of the automorphism space Diff(M), PL(M), or TOP(M) in the smooth, PL, or topological category, respectively, is in any sense known. (As usual, Diff(M) is given the C° ° topology, PL(M) is a simplicial group, and TOP(M) is the singular complex of the homeomorphism group with the compactopen topology.) Besides surgery theory, the principal tool in studying homotopy properties of these automorphism spaces is the concordance space functor C(M) = {automorphisms of M x /fixed on M x 0}. This paper is a survey of some of the main results to date on concordance spaces. Here is an outline of the contents. In §1 we describe how, in a certain stable dimension range, C{M) is a homotopy functor of M, which we denote by ^(M). The application to automorphism spaces is outlined in §2. In §3 we recall the explicit calculations which have been made for %$?(M) and %{g(M), along the lines pioneered by Cerf, and apply them in §4 to compute the group of isotopy classes of automorphisms of the «torus, n ^ 5. §5 is concerned with a stabilized version of #(Af), defined roughly as Q^iS^M), together with the curious equivalence of D^PLOS^M) with ^PL(M)/^Diff(M), due to BurgheleaLashof (based on earlier fundamental work of Morlet). In §6, ^PL(M) is "reduced " to higher simplehomotopy theory. This has some interest in its own right, e.g., it provides a fibered form of Wall's obstruction to finiteness. The important new work of Waldhausen relating <^PL(M) to algebraic ATtheory is outlined, very briefly and imperfectly, in §7. This seems to be the most promising area for future developments in the sub
Stasheff polytopes in algebraic Ktheory and in the space of Morse functions. in Higher homotopy structures in topology and mathematical physics
, 1996
"... 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 multiplicatio ..."
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Cited by 6 (0 self)
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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
Axioms for higher torsion invariants of smooth bundles
, 2005
"... We explain the relationship between various characteristic classes for smooth manifold bundles known as “higher torsion” classes. We isolate two fundamental properties that these cohomology classes may or may not have: additivity and transfer. We show that higher FranzReidemeister torsion and highe ..."
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Cited by 5 (0 self)
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We explain the relationship between various characteristic classes for smooth manifold bundles known as “higher torsion” classes. We isolate two fundamental properties that these cohomology classes may or may not have: additivity and transfer. We show that higher FranzReidemeister torsion and higher MillerMoritaMumford classes satisfy these axioms. Conversely, any characteristic class of smooth bundles satisfying the two axioms must be a linear combination of these two examples. We also show how higher torsion invariants can be computed using only the axioms. Finally, we explain the conjectured formula of S. Goette relating higher analytic torsion classes and higher FranzReidemeister torsion.