Cobordism, homotopy and homology of graphs in r 519 now we define an integer l f by lf i ex, y lf xj y xny where the summation is taken over all disjoint edge pairs of g. In this paper, by use of techniques associated to cobordism theory and morse theory,we give a simple proof of poincare conjecture, i. John milnor 1965, lectures on the h cobordism theorem princeton univ. We will only address dimensions 6 and greater, as dimension 5 limits the theorem. We prove that this theory can reproduce chow groups modulo algebraic equivalence and the semitopological k 0groups. An introduction to cobordism martin vito cruz 30 april 2004 1 introduction cobordism theory is the study of manifolds modulo the cobordism relation. X,y be an s scobordism such that the s action on w is semifree. The princeton legacy library uses the latest printondemand technology to again make available previously outofprint books from the distinguished backlist of princeton university press. The poincar conjecture download ebook pdf, epub, tuebl, mobi. Instead we use the fact that m exists as the boundary of a 6dimensional manifold w. The result is an algebraic version of a theorem due to conner and floyd.
The nilpotent co bration lemma 108 appendices 111 appendix a. Thus the cobordism invariance of the index follows from theorem 1. Roushon these are the notes of some lectures given in the advanced training in mathematics atm workshop on surgery theory and classification of manifolds, june 17 to july 26, 20, northeastern hill. For other interesting applications, the reader is referred to roes paper 6. This may seem like a strange thing to study, but there. I am planning to organize a seminar on cobordism theory and im looking for a reference. This paper begins with an introduction to some of the concepts of algebraic topology.
The h cobordism theorem states that w is diffeomorphic to a product cobordism. A generalization of the hcobordism theorem for possibly non simply connected manifolds is the so called scobordism theorem. M need not be di eomorphic to sn, however, as discussed in an earlier lecture. The audience is familiar with characteristic classes at the level of milnor stasheff. Theorem 1 may be viewed as a geometric presentation of x via. If i may i wish to add to the great list of suggestions the followings. They begin with basics about higher signatures, whitehead torsion and the scobordism theorem. The second half covers some more recent topics, leading to the galatiusmadsentillmannweiss theorem and the cobordism hypothesis.
Lectures on the hcobordism theorem princeton legacy library. Their proof relies on three of their theorems which have had a striking impact on topological manifold theory in general. Seminar on the hcobordism theo rem preliminary list of talks this is a mathematical seminar, for which learning mathematical content and learning to present material that has been extracted from the literature through selfstudy in a form appropriate to the given. In particular, the left hand side is a ring cohomology theory. Stephen smale, 1962, on the structure of manifolds, amer.
A crucial fact used in the proof is due independently to h. On the relation of voevodskys algebraic cobordism to. Pdf the hcobordism theorem is a noted theorem in differential and pl topology. Since considering the kolob theorem my mind has not rested.
First, well consider an object that well call the group of cellular nchains of a. The h cobordism theorem gives sufficient conditions for an h cobordism to be trivial, i. The s cobordism theorem asserts that if dim w 2 6, then an s cobordism is a product. This process is experimental and the keywords may be updated as the learning algorithm improves. The wonder of it all is that the plan of salvation is set before us in the night sky.
Such a reference is preferably a book, but im open to other ideas. No one has succeeded in removing the restriction that v and v have dimension 4. Pdf the equivariant topological scobordism theorem. An scobordism theorem for semifree s1manifolds sciencedirect. The inclusion map of the red subset into the entire space is 1connected. Lectures on the hcobordism theorem 1965 edition open library. We will conclude by proving the poincar e conjecture in high dimensions. The hcobordism theorem showed that simply connected manifolds of dimension at. Get your kindle here, or download a free kindle reading app. The first extension of the h cobordism theorem was given by b. The poincar e conjecture and the hcobordism theo rem the generalized poincar e conjecture is now a theorem.
Smooth scobordisms of elliptic 3manifolds chen, weimin, journal of differential geometry, 2006. Fundamental properties of algebraic cobordism 41 3. However the first author observes that its 5dimensional version is false if the 4dimensional top s cobordism theorem is true. We also show that with finite coefficients, this theory agrees with the algebraic cobordism theory. But it will remain true if we at the same time assume that the inclusion of v. When is a composition cc of an elementary cobordism of. Stong mathematical notes, princeton university press 1968 a detailed table of contents compiled by peter landweber and doug ravenel in november, 2007.
The complex cobordism of bson inoue, koichi and yagita, nobuaki, kyoto journal of mathematics, 2010. Book recommendation for cobordism theory mathoverflow. Milnor, lectures on the hcobordism theorem, notes by l. These keywords were added by machine and not by the authors. Throughout this paper we primarily follow 1 and 2, and most proofs are borrowed from them in some capacity. Milnor, lectures on the h cobordism theorem, notes by l. Theorem 2 is a bad theorem, because it does not generalize to the case. The s cobordism theorem asserts that if dim w 2 6, then an s cobordism. In mathematics, cobordism is a fundamental equivalence relation on the class of compact manifolds of the same dimension, set up using the concept of the boundary french bord, giving cobordism of a manifold. The s cobordism theorem here is where our digression will make sense. Conversely, given a handle decomposition of a cobordism, it comes from a suitable morse function. A classical technique to prove the h cobordism theorem thesis directed by prof.
The s cobordism theorem we have the h cobordism theorem to classify homotopy cobordisms with trivial fundamental group. An explicit isomorphism between morse homology and singular homology is constructed via the technique of pseudocycles. In this paper we will follow a classical technique developed by john milnor in his lectures on the h cobordism theorems half a century ago. These lecture notes contain a guided tour to the novikov conjecture and related conjectures due to baumconnes, borel and farrelljones. Sondow princeton mathematical notes, oxford university press, 18s. The topological scobordism theorem in dimension four or five, proc. The equivariant parametrized hcobordism theorem, the nonmanifold part. These physical invariants often turn out to depend only on the cobordism class of m. The next result is due to barden, mazur, stallings, see 1, 7.
Themanifoldpart we write diff for the category of c. Bulletin new series of the american mathematical society. If we omit the restriction that v and hence v be simply connected, the theorem becomes false. The s cobordism theorem for n 4, let m be a connected, closed nmanifold. Download fulltext pdf the topological s cobordism theorem fails in dimension 4 or 5 article pdf available in mathematical proceedings of the cambridge philosophical society 8401. The goal of this work is the construction of the analogue to the adams spectral sequence in cobordism theory, calculation of the ring of cohomology operations in this theory, and also a number of applications. Pdf the topological scobordism theorem fails in dimension. Ihis theorem has numerous important applications including the proof of the.
Based on the algebraic cobordism theory of levine and morel, we develop a theory of algebraic cobordism modulo algebraic equivalence. The h cobordism theorem showed that simply connected manifolds of dimension at least 5 are much easier than those of dimension 3 or 4. We will now extend this to cases with nontrivial fundamental group. Z m many interesting geometric constructions give results that are only wellde. The morsesmale theorem states that for a morse function on a cobordism, the flowlines of f. This set gives new proof of the h cobordism theorem that is different from the original proof presented by s. Dec 14, 2009 lectures on the h cobordism theorem by john willard milnor, 1965, princeton university press edition, in english.
Jul 04, 2007 pdf file 291 kb article info and citation. The h cobordism theorem gives sufficient conditions for an h cobordism to be equivalent to a cylinder related entries. The statement and consequences of the s cobordism theorem. The generalized poincar e conjecture using cobordism. The h cobordism theorem gives sufficient conditions for an h cobordism to be equivalent to a cylinder. The proof of the theorem depends on the whitney trick of hassler whitney, which geometrically untangles homologicallytangled spheres of complementary dimension in a manifold of dimension 4. The poincar e conjecture and the hcobordism theorem. A classical technique to prove the hcobordism theorem. Numerous examples have been considered and hence a vast literature exists, with few really central theoretical tools, largely due to the idiosyncrasies inherent in the examples. The construction of a sphere s2 from a single d0, a single d1 with its boundary attached to the d0, and two d2 with their boundaries attached to the d1. The book description for lectures on the hcobordism theorem is currently unavailable. We generically write cat for any one of these geometric categories. An introduction to cobordism theory tom weston contents 1. The h cobordism theorem states that an ndim simply connected cobordism w between 2 simply connected n1dim manifolds v,v that are each homotopy equivalent to w which is the definition of an h cobordism is isomorphic to a product, with the isomorphism depending on the category of manifolds with which one is working.
Lectures on the hcobordism theo rem by john willard milnor, 1965, princeton university press edition, in english. Moreover both theories are oriented and the isomorphism above respects the orientations. The proof of the theorem depends on the whitney trick of hassler whitney, which. A generalization of the h cobordism theorem for possibly non simply connected manifolds is the so called s cobordism theorem. That theorem reconstructs complex ktheory via complex cobordism. Lectures on the hcobordism theorem 1965 edition open.
Pdf on feb 1, 1988, mark steinberger and others published the equivariant topological s cobordism theorem find, read and cite all the research you need on researchgate. In this paper, we prove semialgebraic and nash versions of these theorems. This phenomenon is closely related to stokess theorem. The h cobordism theorem is a noted theorem in differential and pl topology. This set gives new proof of the hcobordism theorem that is different from the original proof. 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 pseudocycle and a welldefined integral homology. An application of the whitney embedding theorem and tubular maps is the following theorem. The following adiabatic cobordism theorem for the analytic torsion is an easy corollary of theorem 0. X equal to the disjoint sum of two closed manifolds y 1 and y 2. The basic objects that we will be working with in the proof of the theorem are called cobordisms. Download fulltext pdf the topological scobordism theorem fails in dimension 4 or 5 article pdf available in mathematical proceedings of the cambridge philosophical society 8401. Let x be an even dimensional compact riemannian manifold with boundary.
Adiabatic cobordism theorems for analytic torsion and invariant. That is, starting with semialgebraic or nash cobordism data, we get. Since cobordism theory is a classificational tool, the interest really lies in the investigation of specific classification problems. Lectures on the hcobordism theorem princeton legacy. Let mbe a smooth compact manifold without boundary, of dimension n 5. Suppose that w, v, and v are all simply connected, and the homology group of the pair w, v is trivial.
Controlled simple homotopy theory and applications. The main result that will be proven is the h cobordism theorem, a powerful tool for manifolds of dimension 5 or greater. The topological scobordism theorem fails in dimension 4 or 5. Pdf semialgebraic hcobordism and scobordism theorems. The equivariant parametrized h cobordism theorem, the nonmanifold part. Citeseerx citation query lectures on the hcobordism theorem.