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

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 Grobman-Hartman theorem and the λ-lemma (Inclination Lemma) to analyze compactness and define gluing for the moduli space of flow lines.

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.

We give an interpretation of Yetter’s Invariant of manifolds M in terms of the homotopy type of the function space TOP(M,B(G)), where G is a crossed module and B(G) is its classifying space. From this formulation, there follows that Yetter’s invariant depends only on the homotopy type of M, and the weak homotopy type of the crossed module G. We use this interpretation to define a twisting of Yetter’s Invariant by cohomology classes of crossed modules, defined

Homotopy Quantum Field Theories (HQFTs) were introduced by the second author to extend the ideas and methods of Topological Quantum Field Theories to closed d-manifolds 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 2-type, giving a common generalisation of the classifying algebraic structures for the two cases previously known. The algebraic models for 2-types that we use are crossed modules, C, and we introduce a notion of formal C-map, which extends the usual lattice-type constructions to this setting. This leads to a classification of ‘formal ’ 2-dimensional HQFTs with target C,

The 4-dimensional 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 disk-embedding theorem in the simply-connected setting, proved in [3].) For example, this may be done when the surgery kernel is represented by π1-null spheres [4], or more generally by a π1-null 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 4-dimensional 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 4-manifold. 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 4-manifold with free fundamental group, and suppose the intersection form on M is extended from the integers. Then M is s-cobordant to a connected sum of ♯ n S 1 × S 3 with a simply-connected 4-manifold. Note that if the surgery conjecture fails for free groups, then for both theorems above there is, in general, no extension to 4-manifolds with boundary. The assumption on the intersection pairing in theorem 2 is necessary, since there are forms

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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Categorifying the concept of topological group, one obtains the notion of a ‘topological 2-group’. This in turn allows a theory of ‘principal 2-bundles’ generalizing the usual theory of principal bundles. It is well-known that under mild conditions on a topological group G and a space M, principal G-bundles over M are classified by either the Čech cohomology ˇ H 1 (M, G) or the set of homotopy classes [M, BG], where BG is the classifying space of G. Here we review work by Bartels, Jurčo, Baas–Bökstedt–Kro, and others generalizing this result to topological 2-groups and even topological 2-categories. We explain various viewpoints on topological 2-groups and the Čech cohomology ˇ H 1 (M, G) with coefficients in a topological 2-group G, also known as ‘nonabelian cohomology’. Then we give an elementary proof that under mild conditions on M and G there is a bijection ˇH 1 (M, G) ∼ = [M, B|G|] where B|G | is the classifying space of the geometric realization of the nerve of G. Applying this result to the ‘string 2-group ’ String(G) of a simply-connected compact simple Lie group G, it follows that principal String(G)-2-bundles have rational characteristic classes coming from elements of H ∗ (BG, Q)/〈c〉, where c is any generator of H 4 (BG, Q).

We explore the relations among quadratic modules, 2-crossed modules, crossed squares and simplicial groups with Moore complex of length 2.