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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 HARE AND THE TORTOISE
"... It is a pleasure to be here to help celebrate Mike Boardman’s 60th birthday. 1 I have just finished writing a history of stable algebraic topology from the end of World War II through 1966 [18]. The starting point was natural enough. The paper of Eilenberg and Mac Lane [6] that introduced the catego ..."
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It is a pleasure to be here to help celebrate Mike Boardman’s 60th birthday. 1 I have just finished writing a history of stable algebraic topology from the end of World War II through 1966 [18]. The starting point was natural enough. The paper of Eilenberg and Mac Lane [6] that introduced the categorical language we now all speak appeared in 1945, and so did the paper of Eilenberg and Steenrod [7] that announced the axiomatic treatment of homology and cohomology. The ending point was more artificial, at first dictated by constraints of time and energy and the fact that Steenrod’s compendium of Mathematical Reviews in topology contained all reviews published through 1967 and thus all papers published through 1966. It also made it easy for me to be modest and impersonal. Although I got my PhD in 1964, I only plugged into the circuit and began to know what was going on when I arrived at Chicago, at the end of 1966. Mike also got his PhD in 1964. Since he is two years older than I am, I guess he was a little slow. But then, his thesis was a lot more important than mine was, although people at the time didn’t seem to understand that. Its results became available in a mimeographed summary in 1966. So maybe 1966 wasn’t such a bad stopping point mathematically. It is amazing how much we didn’t know then, how many familiar names had not yet made their mark. In fact, a complete list of the people who made sustained and important contributions to the development of stable algebraic topology in the years 1945 through 1966 would have no more than around 40 names on it. On the other hand, the caliber of the people working in the field was extraordinary.
Printed in the United States of America Library of Congress Cataloging in Publication Data
"... will be found on the last printed page of this book. Preface The text which follows is based mostly on lectures at Princeton University in 1957. The senior author wishes to apologize for the delay in publication. The theory of characteristic classes began in the year 1935 with almost simultaneous wo ..."
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will be found on the last printed page of this book. Preface The text which follows is based mostly on lectures at Princeton University in 1957. The senior author wishes to apologize for the delay in publication. The theory of characteristic classes began in the year 1935 with almost simultaneous work by HASSLER WHITNEY in the United States and EDUARD STIEFEL in Switzerland. Stiefe1's thesis, written under the direction of Heinz Hopf, introduced and studied certain "characteristic" homology classes determined by the tangent bundle of a smooth manifold. Whitney, then at Harvard University, treated the case of an arbitrary sphere bundle. Somewhat later he invented the language of cohomo10gy theory, hence the concept of a characteristic cohomology class, and proved the basic product theorem. In 1942 LEV PONTRJAGIN of Moscow University began to study the homology of Grassmann manifolds, using a cell subdivision due to Charles Ehresmann. This enabled him to construct important new characteristic classes. (Pontrjagin's many contributions to mathematics are the more remarkable in that he is totally blind, having lost his eyesight in an accident at the age of fourteen.) In 1946 SHINGSHEN CHERN, recently arrived at the Institute for Advanced Study from Kunming in southwestern China, defined characteri:stic classes for complex vector bundles. In fact he showed that the complex Grassmann manifolds have a cohomology structure which is much easier to understand than that of the real Grassmann manifolds. This has led to a great clarification of the theory of real characteristic classes, v vi