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̂HF(Y, s),and HFred(Y, s) associated to closed, oriented three-manifolds Y equipped with a Spin c structures s ∈ Spin c (Y). In the present paper, we give calculations and study the properties of these invariants. The calculations suggest a conjectured relationship with Seiberg-Witten theory. The properties include a relationship between the Euler characteristics of HF ± and Turaev’s torsion, a relationship with the minimal genus problem (Thurston norm), and surgery exact sequences. We also include some applications of these techniques to three-manifold topology. 1.

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Abstract. An explicit isomorphism between Morse homology and singular homology is constructed via the technique of pseudo-cycles. Given a Morse cycle as a formal sum of critical points of a Morse function, the unstable manifolds for the negative gradient flow are compactified in a suitable way, such that gluing them appropriately leads to a pseudo-cycle and a well-defined integral homology class in singular homology. 1.

Given a compact Riemannianmanifold (M d ; g), a finite dimensional representation ae : 1 (M) ! GL(V ) of the fundamental group 1 (M) on a vectorspace V of dimension l and a Hermitian structure on the flat vector bundle E p !M associated to ae, Ray-Singer [RS] have introduced the analytic torsion T = T (M; ae; g; ) ? 0: Witten's deformation dq (t) of the exterior derivative dq ; dq (t) = e \Gammaht dq e ht ; with h : M ! R a smooth Morse function, can be used to define a deformation T (h; t) ? 0 of the analytic torsion T with T (h; 0) = T: The main results of this paper are to provide, assuming that grad g h is Morse Smale, an asymptotic expansion for log T (h; t) for t ! 1 of the form P d+1 j=0 a j t j + b log t + O( 1 p t ) and to present two different formulae for a 0 : As an application we obtain a shorter derivation of results due to Ray-Singer [RS], Cheeger [Ch], Muller [Mu1,2] and Bismut-Zhang [BZ] which, in increasing generality, concern the equality for ...

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Abstract. For manifolds with freely generated first homology, we characterize gerbes with connection as functors on a certain surface cobordism category. This allows us to relate gerbes with connection to Turaev’s 1+1-dimensional homotopy quantum field theories, and we show that flat gerbes on such spaces as above are the same as a specific class of rank one homotopy quantum field theories.

Abstract. We outline the current state of knowledge regarding geometric inequalities of systolic type, and prove new results, including systolic freedom in dimension 4. Namely, every compact, orientable, smooth 4-manifold X admits metrics of arbitrarily small volume such that every orientable, immersed surface of smaller than unit area is necessarily null-homologous in X. In other words, orientable 4-manifolds are 2-systolically free. More generally, let m be a positive even integer, and let n> m. Then all manifolds of dimension at most n are m-systolically free (modulo torsion) if all k-skeleta, m+1 ≤ k ≤ n, of the loop space Ω(S m+1) are m-systolically free. 1.