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22
Elliptic operators on manifolds with singularities and Khomology
, 2008
"... Elliptic operators on smooth compact manifolds are classified by Khomology. We prove that a similar classification is valid also for manifolds with simplest singularities: isolated conical points and edges. The main ingredients of the proof of these results are: AtiyahSinger difference constructio ..."
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Elliptic operators on smooth compact manifolds are classified by Khomology. We prove that a similar classification is valid also for manifolds with simplest singularities: isolated conical points and edges. The main ingredients of the proof of these results are: AtiyahSinger difference construction in the noncommutative case and Poincare isomorphism in Ktheory for (our) singular manifolds. As an application we give a formula in topological terms for the obstruction to Fredholm problems on manifolds with edges.
The Teichmüller Space of Pinched Negatively Curved Metrics on a Hyperbolic Manifold is not Contractible
, 2006
"... For a smooth manifold M we define the Teichmüller space T (M) of all Riemannian metrics on M and the Teichmüller space T ǫ (M) of ǫpinched negatively curved metrics on M, where 0 ≤ ǫ ≤ ∞. We prove that if M is hyperbolic the natural inclusion ..."
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For a smooth manifold M we define the Teichmüller space T (M) of all Riemannian metrics on M and the Teichmüller space T ǫ (M) of ǫpinched negatively curved metrics on M, where 0 ≤ ǫ ≤ ∞. We prove that if M is hyperbolic the natural inclusion
LOOP PRODUCTS AND CLOSED GEODESICS
"... Abstract. The critical points of the length function on the free loop space Λ(M) of a compact Riemannian manifold M are the closed geodesics on M. The length function gives a filtration of the homology of Λ(M) and we show that the ChasSullivan product Hi(Λ) × Hj(Λ) ∗ ✲ Hi+j−n(Λ) is compatible wi ..."
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Abstract. The critical points of the length function on the free loop space Λ(M) of a compact Riemannian manifold M are the closed geodesics on M. The length function gives a filtration of the homology of Λ(M) and we show that the ChasSullivan product Hi(Λ) × Hj(Λ) ∗ ✲ Hi+j−n(Λ) is compatible with this filtration. We obtain a very simple expression for the associated graded homology ring GrH∗(Λ(M)) when all geodesics are closed, or when all geodesics are nondegenerate. We also interpret Sullivan’s coproduct ∨ [Su1, Su2] on C∗(Λ) as a product in cohomology
A History of Duality in Algebraic Topology
"... This paper became the starting point of investigations of homology for more general spaces than merely finite complexes or open subsets of R ..."
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This paper became the starting point of investigations of homology for more general spaces than merely finite complexes or open subsets of R
THE ARFKERVAIRE INVARIANT OF FRAMED MANIFOLDS
"... 2. Stable homotopy groups of spheres 3 3. Framed manifolds and stable homotopy groups 6 ..."
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2. Stable homotopy groups of spheres 3 3. Framed manifolds and stable homotopy groups 6