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67
Spaces of maps into classifying spaces for equivariant crossed complexes
 Indag. Math. (N.S
, 1997
"... Abstract. The results of a previous paper on the equivariant homotopy theory of crossed complexes are generalised from the case of a discrete group to general topological groups. The principal new ingredient necessary for this is an analysis of homotopy coherence theory for crossed complexes, using ..."
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Abstract. The results of a previous paper on the equivariant homotopy theory of crossed complexes are generalised from the case of a discrete group to general topological groups. The principal new ingredient necessary for this is an analysis of homotopy coherence theory for crossed complexes, using detailed results on the appropriate Eilenberg–Zilber theory, and of its relation to simplicial homotopy coherence. Again, our results give information not just on the homotopy classification of certain equivariant maps, but also on the weak equivariant homotopy type of the corresponding equivariant function spaces. Mathematics Subject Classifications (2001): 55P91, 55U10, 18G55. Key words: equivariant homotopy theory, classifying space, function space, crossed complex.
Computations and homotopical applications of induced crossed modules
 J. Symb. Comp
"... We explain how the computation of induced crossed modules allows the computation of certain homotopy 2types and, in particular, second homotopy groups. We discuss various issues involved in computing induced crossed modules and give some examples and applications. ..."
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Cited by 10 (8 self)
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We explain how the computation of induced crossed modules allows the computation of certain homotopy 2types and, in particular, second homotopy groups. We discuss various issues involved in computing induced crossed modules and give some examples and applications.
Computing homotopy types using crossed ncubes of groups
 in Adams Memorial Symposium on Algebraic Topology
, 1992
"... Dedicated to the memory of Frank Adams ..."
The MorseWitten complex via dynamical systems
 Expo. Math
"... to a Morse function f and a Riemannian metric g on M consists of chain groups generated by the critical points of f and a boundary operator counting isolated flow lines of the negative gradient flow. Its homology reproduces singular homology of M. The geometric approach presented here was developed ..."
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Cited by 6 (1 self)
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to a Morse function f and a Riemannian metric g on M consists of chain groups generated by the critical points of f and a boundary operator counting isolated flow lines of the negative gradient flow. Its homology reproduces singular homology of M. The geometric approach presented here was developed in [We93] and is based on tools from hyperbolic dynamical systems. For instance, we apply the GrobmanHartman theorem and the λlemma (Inclination Lemma) to analyze compactness and define gluing for the moduli space of flow lines.
Surgery on closed 4manifolds with free fundamental group
 Math. Proc. Cambridge Philos. Soc
"... The 4dimensional topological surgery conjecture has been established for a class of groups, including the groups of subexponential growth (see [6], [13] for recent developments), however the general case remains open. The full surgery conjecture is known to be equivalent to the question for a class ..."
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Cited by 6 (2 self)
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The 4dimensional topological surgery conjecture has been established for a class of groups, including the groups of subexponential growth (see [6], [13] for recent developments), however the general case remains open. The full surgery conjecture is known to be equivalent to the question for a class of canonical problems with free fundamental group [5, Chapter 12]. The proof of the conjecture for “good ” groups relies on the disk embedding theorem (see [5]), which is not presently known to hold for arbitrary groups. However, in certain cases it may be shown that surgery works even when the disk embedding theorem is not available for a given fundamental group (such results still use the diskembedding theorem in the simplyconnected setting, proved in [3].) For example, this may be done when the surgery kernel is represented by π1null spheres [4], or more generally by a π1null submanifold satisfying a certain condition on Dwyer’s filtration on second homology [7]. Here we state another instance when the surgery conjecture holds for free groups. The following results are stated in the topological category. Theorem 1. Let X be a 4dimensional Poincaré complex with free fundamental group, and assume the intersection form on X is extended from the integers. Let f: M − → X be a degree 1 normal map, where M is a closed 4manifold. Then the vanishing of the Wall obstruction implies that f is normally bordant to a homotopy equivalence f ′ : M ′ − → X. In the canonical surgery problems, X has free fundamental group and trivial π2, however what makes them harder to analyze is the interplay between the homotopy type of X, and the topology of the boundary. Our result sidesteps this by considering closed manifolds. We also prove a related splitting result: Theorem 2. Let M be a closed orientable 4manifold with free fundamental group, and suppose the intersection form on M is extended from the integers. Then M is scobordant to a connected sum of ♯ n S 1 × S 3 with a simplyconnected 4manifold. Note that if the surgery conjecture fails for free groups, then for both theorems above there is, in general, no extension to 4manifolds with boundary. The assumption on the intersection pairing in theorem 2 is necessary, since there are forms
Galois Theory of Second Order Covering Maps of Simplicial Sets
 J. Pure Appl. Algebra
, 1995
"... this paper is to develop such a theory for simplicial sets, as a special case of Galois theory in categories [6]. The second order notion of fundamental groupoid arising here as the Galois groupoid of a fibration is slightly different from the above notions but ..."
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Cited by 6 (3 self)
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this paper is to develop such a theory for simplicial sets, as a special case of Galois theory in categories [6]. The second order notion of fundamental groupoid arising here as the Galois groupoid of a fibration is slightly different from the above notions but
The Proportionality Principle of Simplicial Volume
, 2004
"... Manifolds, the basic objects of geometry, form the playground for both topological and smooth structures. Many challenges in modern mathematics are concerned with the nature of this interaction between algebraic topology, differential topology and differential geometry. For example all incarnations ..."
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Cited by 6 (0 self)
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Manifolds, the basic objects of geometry, form the playground for both topological and smooth structures. Many challenges in modern mathematics are concerned with the nature of this interaction between algebraic topology, differential topology and differential geometry. For example all incarnations
On a General Chain Model of the Free Loop Space And String Topology
, 2007
"... Let M be a smooth oriented manifold. The homology of M has the structure of a Frobenius algebra. This paper shows that on chain level there is a Frobeniuslike algebra structure, whose homology gives the Frobenius algebra of M. Moreover, associated to any Frobeniuslike algebra, there is a chain comp ..."
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Cited by 5 (1 self)
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Let M be a smooth oriented manifold. The homology of M has the structure of a Frobenius algebra. This paper shows that on chain level there is a Frobeniuslike algebra structure, whose homology gives the Frobenius algebra of M. Moreover, associated to any Frobeniuslike algebra, there is a chain complex whose homology has the structure of a Gerstenhaber algebra and a BatalinVilkovisky algebra. And if the Frobeniuslike algebra comes from M, it gives the free loop space LM and String Topology of ChasSullivan. Contents
Secondary homotopy groups
 Preprint of the MaxPlanckInstitut für Mathematik MPIM200636, http://arxiv.org/abs/math.AT/0604029
, 2005
"... Abstract. Secondary homotopy groups supplement the structure of classical homotopy groups. They yield a track functor on the track category of pointed spaces compatible with fiber sequences, suspensions and loop spaces. They also yield algebraic models of (n − 1)connected (n + 1)types for n ≥ 0. ..."
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Cited by 5 (5 self)
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Abstract. Secondary homotopy groups supplement the structure of classical homotopy groups. They yield a track functor on the track category of pointed spaces compatible with fiber sequences, suspensions and loop spaces. They also yield algebraic models of (n − 1)connected (n + 1)types for n ≥ 0.
Formal Homotopy Quantum Field Theories
 II : Simplicial Formal Maps
"... Homotopy Quantum Field Theories (HQFTs) were introduced by the second author to extend the ideas and methods of Topological Quantum Field Theories to closed dmanifolds endowed with extra structure in the form of homotopy classes of maps into a given ‘target ’ space. For d = 1, classifications of HQ ..."
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Cited by 5 (2 self)
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Homotopy Quantum Field Theories (HQFTs) were introduced by the second author to extend the ideas and methods of Topological Quantum Field Theories to closed dmanifolds endowed with extra structure in the form of homotopy classes of maps into a given ‘target ’ space. For d = 1, classifications of HQFTs in terms of algebraic structures are known when B is a K(G,1) and also when it is simply connected. Here we study general HQFTs with d = 1 and target a general 2type, giving a common generalisation of the classifying algebraic structures for the two cases previously known. The algebraic models for 2types that we use are crossed modules, C, and we introduce a notion of formal Cmap, which extends the usual latticetype constructions to this setting. This leads to a classification of ‘formal ’ 2dimensional HQFTs with target C,