Toric Polynomial Generators of Complex Cobordism
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HE WANG Abstract. a Mini-Course on Rational Homotopy Theory
RATIONAL HOMOTOPY THEORY HE WANG Abstract. A mini-course on rational homotopy theory. Contents 1. Introduction 2 2. Elementary homotopy theory 3 3. Spectral sequences 8 4. Postnikov towers and rational homotopy theory 16 5. Commutative differential graded algebras 21 6. Minimal models 25 7. Fundamental groups 34 References 36 2010 Mathematics Subject Classification. Primary 55P62 . 1 2 HE WANG 1. Introduction One of the goals of topology is to classify the topological spaces up to some equiva- lence relations, e.g., homeomorphic equivalence and homotopy equivalence (for algebraic topology). In algebraic topology, most of the time we will restrict to spaces which are homotopy equivalent to CW complexes. We have learned several algebraic invariants such as fundamental groups, homology groups, cohomology groups and cohomology rings. Using these algebraic invariants, we can seperate two non-homotopy equivalent spaces. Another powerful algebraic invariants are the higher homotopy groups. Whitehead the- orem shows that the functor of homotopy theory are power enough to determine when two CW complex are homotopy equivalent. A rational coefficient version of the homotopy theory has its own techniques and advan- tages: 1. fruitful algebraic structures. 2. easy to calculate. RATIONAL HOMOTOPY THEORY 3 2. Elementary homotopy theory 2.1. Higher homotopy groups. Let X be a connected CW-complex with a base point x0. Recall that the fundamental group π1(X; x0) = [(I;@I); (X; x0)] is the set of homotopy classes of maps from pair (I;@I) to (X; x0) with the product defined by composition of paths. Similarly, for each n ≥ 2, the higher homotopy group n n πn(X; x0) = [(I ;@I ); (X; x0)] n n is the set of homotopy classes of maps from pair (I ;@I ) to (X; x0) with the product defined by composition. -
An Introduction to Complex Algebraic Geometry with Emphasis on The
AN INTRODUCTION TO COMPLEX ALGEBRAIC GEOMETRY WITH EMPHASIS ON THE THEORY OF SURFACES By Chris Peters Mathematisch Instituut der Rijksuniversiteit Leiden and Institut Fourier Grenoble i Preface These notes are based on courses given in the fall of 1992 at the University of Leiden and in the spring of 1993 at the University of Grenoble. These courses were meant to elucidate the Mori point of view on classification theory of algebraic surfaces as briefly alluded to in [P]. The material presented here consists of a more or less self-contained advanced course in complex algebraic geometry presupposing only some familiarity with the theory of algebraic curves or Riemann surfaces. But the goal, as in the lectures, is to understand the Enriques classification of surfaces from the point of view of Mori-theory. In my opininion any serious student in algebraic geometry should be acquainted as soon as possible with the yoga of coherent sheaves and so, after recalling the basic concepts in algebraic geometry, I have treated sheaves and their cohomology theory. This part culminated in Serre’s theorems about coherent sheaves on projective space. Having mastered these tools, the student can really start with surface theory, in particular with intersection theory of divisors on surfaces. The treatment given is algebraic, but the relation with the topological intersection theory is commented on briefly. A fuller discussion can be found in Appendix 2. Intersection theory then is applied immediately to rational surfaces. A basic tool from the modern point of view is Mori’s rationality theorem. The treatment for surfaces is elementary and I borrowed it from [Wi]. -
The Relation of Cobordism to K-Theories
The Relation of Cobordism to K-Theories PhD student seminar winter term 2006/07 November 7, 2006 In the current winter term 2006/07 we want to learn something about the different flavours of cobordism theory - about its geometric constructions and highly algebraic calculations using spectral sequences. Further we want to learn the modern language of Thom spectra and survey some levels in the tower of the following Thom spectra MO MSO MSpin : : : Mfeg: After that there should be time for various special topics: we could calculate the homology or homotopy of Thom spectra, have a look at the theorem of Hartori-Stong, care about d-,e- and f-invariants, look at K-, KO- and MU-orientations and last but not least prove some theorems of Conner-Floyd type: ∼ MU∗X ⊗MU∗ K∗ = K∗X: Unfortunately, there is not a single book that covers all of this stuff or at least a main part of it. But on the other hand it is not a big problem to use several books at a time. As a motivating introduction I highly recommend the slides of Neil Strickland. The 9 slides contain a lot of pictures, it is fun reading them! • November 2nd, 2006: Talk 1 by Marc Siegmund: This should be an introduction to bordism, tell us the G geometric constructions and mention the ring structure of Ω∗ . You might take the books of Switzer (chapter 12) and Br¨oker/tom Dieck (chapter 2). Talk 2 by Julia Singer: The Pontrjagin-Thom construction should be given in detail. Have a look at the books of Hirsch, Switzer (chapter 12) or Br¨oker/tom Dieck (chapter 3). -
Algebraic Cobordism
Algebraic Cobordism Marc Levine January 22, 2009 Marc Levine Algebraic Cobordism Outline I Describe \oriented cohomology of smooth algebraic varieties" I Recall the fundamental properties of complex cobordism I Describe the fundamental properties of algebraic cobordism I Sketch the construction of algebraic cobordism I Give an application to Donaldson-Thomas invariants Marc Levine Algebraic Cobordism Algebraic topology and algebraic geometry Marc Levine Algebraic Cobordism Algebraic topology and algebraic geometry Naive algebraic analogs: Algebraic topology Algebraic geometry ∗ ∗ Singular homology H (X ; Z) $ Chow ring CH (X ) ∗ alg Topological K-theory Ktop(X ) $ Grothendieck group K0 (X ) Complex cobordism MU∗(X ) $ Algebraic cobordism Ω∗(X ) Marc Levine Algebraic Cobordism Algebraic topology and algebraic geometry Refined algebraic analogs: Algebraic topology Algebraic geometry The stable homotopy $ The motivic stable homotopy category SH category over k, SH(k) ∗ ∗;∗ Singular homology H (X ; Z) $ Motivic cohomology H (X ; Z) ∗ alg Topological K-theory Ktop(X ) $ Algebraic K-theory K∗ (X ) Complex cobordism MU∗(X ) $ Algebraic cobordism MGL∗;∗(X ) Marc Levine Algebraic Cobordism Cobordism and oriented cohomology Marc Levine Algebraic Cobordism Cobordism and oriented cohomology Complex cobordism is special Complex cobordism MU∗ is distinguished as the universal C-oriented cohomology theory on differentiable manifolds. We approach algebraic cobordism by defining oriented cohomology of smooth algebraic varieties, and constructing algebraic cobordism as the universal oriented cohomology theory. Marc Levine Algebraic Cobordism Cobordism and oriented cohomology Oriented cohomology What should \oriented cohomology of smooth varieties" be? Follow complex cobordism MU∗ as a model: k: a field. Sm=k: smooth quasi-projective varieties over k. An oriented cohomology theory A on Sm=k consists of: D1. -
Circles in a Complex Projective Space
Adachi, T., Maeda, S. and Udagawa, S. Osaka J. Math. 32 (1995), 709-719 CIRCLES IN A COMPLEX PROJECTIVE SPACE TOSHIAKI ADACHI, SADAHIRO MAEDA AND SEIICHI UDAGAWA (Received February 15, 1994) 0. Introduction The study of circles is one of the interesting objects in differential geometry. A curve γ(s) on a Riemannian manifold M parametrized by its arc length 5 is called a circle, if there exists a field of unit vectors Ys along the curve which satisfies, together with the unit tangent vectors Xs — Ϋ(s)9 the differential equations : FSXS = kYs and FsYs= — kXs, where k is a positive constant, which is called the curvature of the circle γ(s) and Vs denotes the covariant differentiation along γ(s) with respect to the Aiemannian connection V of M. For given a point lEJIί, orthonormal pair of vectors u, v^ TXM and for any given positive constant k, we have a unique circle γ(s) such that γ(0)=x, γ(0) = u and (Psγ(s))s=o=kv. It is known that in a complete Riemannian manifold every circle can be defined for -oo<5<oo (Cf. [6]). The study of global behaviours of circles is very interesting. However there are few results in this direction except for the global existence theorem. In general, a circle in a Riemannian manifold is not closed. Here we call a circle γ(s) closed if = there exists So with 7(so) = /(0), XSo Xo and YSo— Yo. Of course, any circles in Euclidean m-space Em are closed. And also any circles in Euclidean m-sphere Sm(c) are closed. -
NOTES on CARTIER and WEIL DIVISORS Recall: Definition 0.1. A
NOTES ON CARTIER AND WEIL DIVISORS AKHIL MATHEW Abstract. These are notes on divisors from Ravi Vakil's book [2] on scheme theory that I prepared for the Foundations of Algebraic Geometry seminar at Harvard. Most of it is a rewrite of chapter 15 in Vakil's book, and the originality of these notes lies in the mistakes. I learned some of this from [1] though. Recall: Definition 0.1. A line bundle on a ringed space X (e.g. a scheme) is a locally free sheaf of rank one. The group of isomorphism classes of line bundles is called the Picard group and is denoted Pic(X). Here is a standard source of line bundles. 1. The twisting sheaf 1.1. Twisting in general. Let R be a graded ring, R = R0 ⊕ R1 ⊕ ::: . We have discussed the construction of the scheme ProjR. Let us now briefly explain the following additional construction (which will be covered in more detail tomorrow). L Let M = Mn be a graded R-module. Definition 1.1. We define the sheaf Mf on ProjR as follows. On the basic open set D(f) = SpecR(f) ⊂ ProjR, we consider the sheaf associated to the R(f)-module M(f). It can be checked easily that these sheaves glue on D(f) \ D(g) = D(fg) and become a quasi-coherent sheaf Mf on ProjR. Clearly, the association M ! Mf is a functor from graded R-modules to quasi- coherent sheaves on ProjR. (For R reasonable, it is in fact essentially an equiva- lence, though we shall not need this.) We now set a bit of notation. -
New Ideas in Algebraic Topology (K-Theory and Its Applications)
NEW IDEAS IN ALGEBRAIC TOPOLOGY (K-THEORY AND ITS APPLICATIONS) S.P. NOVIKOV Contents Introduction 1 Chapter I. CLASSICAL CONCEPTS AND RESULTS 2 § 1. The concept of a fibre bundle 2 § 2. A general description of fibre bundles 4 § 3. Operations on fibre bundles 5 Chapter II. CHARACTERISTIC CLASSES AND COBORDISMS 5 § 4. The cohomological invariants of a fibre bundle. The characteristic classes of Stiefel–Whitney, Pontryagin and Chern 5 § 5. The characteristic numbers of Pontryagin, Chern and Stiefel. Cobordisms 7 § 6. The Hirzebruch genera. Theorems of Riemann–Roch type 8 § 7. Bott periodicity 9 § 8. Thom complexes 10 § 9. Notes on the invariance of the classes 10 Chapter III. GENERALIZED COHOMOLOGIES. THE K-FUNCTOR AND THE THEORY OF BORDISMS. MICROBUNDLES. 11 § 10. Generalized cohomologies. Examples. 11 Chapter IV. SOME APPLICATIONS OF THE K- AND J-FUNCTORS AND BORDISM THEORIES 16 § 11. Strict application of K-theory 16 § 12. Simultaneous applications of the K- and J-functors. Cohomology operation in K-theory 17 § 13. Bordism theory 19 APPENDIX 21 The Hirzebruch formula and coverings 21 Some pointers to the literature 22 References 22 Introduction In recent years there has been a widespread development in topology of the so-called generalized homology theories. Of these perhaps the most striking are K-theory and the bordism and cobordism theories. The term homology theory is used here, because these objects, often very different in their geometric meaning, Russian Math. Surveys. Volume 20, Number 3, May–June 1965. Translated by I.R. Porteous. 1 2 S.P. NOVIKOV share many of the properties of ordinary homology and cohomology, the analogy being extremely useful in solving concrete problems. -
Floer Homology, Gauge Theory, and Low-Dimensional Topology
Floer Homology, Gauge Theory, and Low-Dimensional Topology Clay Mathematics Proceedings Volume 5 Floer Homology, Gauge Theory, and Low-Dimensional Topology Proceedings of the Clay Mathematics Institute 2004 Summer School Alfréd Rényi Institute of Mathematics Budapest, Hungary June 5–26, 2004 David A. Ellwood Peter S. Ozsváth András I. Stipsicz Zoltán Szabó Editors American Mathematical Society Clay Mathematics Institute 2000 Mathematics Subject Classification. Primary 57R17, 57R55, 57R57, 57R58, 53D05, 53D40, 57M27, 14J26. The cover illustrates a Kinoshita-Terasaka knot (a knot with trivial Alexander polyno- mial), and two Kauffman states. These states represent the two generators of the Heegaard Floer homology of the knot in its topmost filtration level. The fact that these elements are homologically non-trivial can be used to show that the Seifert genus of this knot is two, a result first proved by David Gabai. Library of Congress Cataloging-in-Publication Data Clay Mathematics Institute. Summer School (2004 : Budapest, Hungary) Floer homology, gauge theory, and low-dimensional topology : proceedings of the Clay Mathe- matics Institute 2004 Summer School, Alfr´ed R´enyi Institute of Mathematics, Budapest, Hungary, June 5–26, 2004 / David A. Ellwood ...[et al.], editors. p. cm. — (Clay mathematics proceedings, ISSN 1534-6455 ; v. 5) ISBN 0-8218-3845-8 (alk. paper) 1. Low-dimensional topology—Congresses. 2. Symplectic geometry—Congresses. 3. Homol- ogy theory—Congresses. 4. Gauge fields (Physics)—Congresses. I. Ellwood, D. (David), 1966– II. Title. III. Series. QA612.14.C55 2004 514.22—dc22 2006042815 Copying and reprinting. Material in this book may be reproduced by any means for educa- tional and scientific purposes without fee or permission with the exception of reproduction by ser- vices that collect fees for delivery of documents and provided that the customary acknowledgment of the source is given. -
On the Bordism Ring of Complex Projective Space Claude Schochet
proceedings of the american mathematical society Volume 37, Number 1, January 1973 ON THE BORDISM RING OF COMPLEX PROJECTIVE SPACE CLAUDE SCHOCHET Abstract. The bordism ring MUt{CPa>) is central to the theory of formal groups as applied by D. Quillen, J. F. Adams, and others recently to complex cobordism. In the present paper, rings Et(CPm) are considered, where E is an oriented ring spectrum, R=7rt(£), andpR=0 for a prime/». It is known that Et(CPcc) is freely generated as an .R-module by elements {ßT\r^0}. The ring structure, however, is not known. It is shown that the elements {/VI^O} form a simple system of generators for £t(CP°°) and that ßlr=s"rß„r mod(/?j, • • • , ßvr-i) for an element s e R (which corre- sponds to [CP"-1] when E=MUZV). This may lead to information concerning Et(K(Z, n)). 1. Introduction. Let E he an associative, commutative ring spectrum with unit, with R=n*E. Then E determines a generalized homology theory £* and a generalized cohomology theory E* (as in G. W. White- head [5]). Following J. F. Adams [1] (and using his notation throughout), assume that E is oriented in the following sense : There is given an element x e £*(CPX) such that £*(S2) is a free R- module on i*(x), where i:S2=CP1-+CP00 is the inclusion. (The Thom-Milnor spectrum MU which yields complex bordism theory and cobordism theory satisfies these hypotheses and is of seminal interest.) By a spectral sequence argument and general nonsense, Adams shows: (1.1) E*(CP ) is the graded ring of formal power series /?[[*]]. -
Grassmannians Via Projection Operators and Some of Their Special Submanifolds ∗
GRASSMANNIANS VIA PROJECTION OPERATORS AND SOME OF THEIR SPECIAL SUBMANIFOLDS ∗ IVKO DIMITRIC´ Department of Mathematics, Penn State University Fayette, Uniontown, PA 15401-0519, USA E-mail: [email protected] We study submanifolds of a general Grassmann manifold which are of 1-type in a suitably defined Euclidean space of F−Hermitian matrices (such a submanifold is minimal in some hypersphere of that space and, apart from a translation, the im- mersion is built using eigenfunctions from a single eigenspace of the Laplacian). We show that a minimal 1-type hypersurface of a Grassmannian is mass-symmetric and has the type-eigenvalue which is a specific number in each dimension. We derive some conditions on the mean curvature of a 1-type hypersurface of a Grassmannian and show that such a hypersurface with a nonzero constant mean curvature has at most three constant principal curvatures. At the end, we give some estimates of the first nonzero eigenvalue of the Laplacian on a compact submanifold of a Grassmannian. 1. Introduction The Grassmann manifold of k dimensional subspaces of the vector space − Fm over a field F R, C, H can be conveniently defined as the base man- ∈{ } ifold of the principal fibre bundle of the corresponding Stiefel manifold of F unitary frames, where a frame is projected onto the F plane spanned − − by it. As observed already in 1927 by Cartan, all Grassmannians are sym- metric spaces and they have been studied in such context ever since, using the root systems and the Lie algebra techniques. However, the submani- fold geometry in Grassmannians of higher rank is much less developed than the corresponding geometry in rank-1 symmetric spaces, the notable excep- tions being the classification of the maximal totally geodesic submanifolds (the well known results of J.A. -
CR Singular Immersions of Complex Projective Spaces
Beitr¨agezur Algebra und Geometrie Contributions to Algebra and Geometry Volume 43 (2002), No. 2, 451-477. CR Singular Immersions of Complex Projective Spaces Adam Coffman∗ Department of Mathematical Sciences Indiana University Purdue University Fort Wayne Fort Wayne, IN 46805-1499 e-mail: Coff[email protected] Abstract. Quadratically parametrized smooth maps from one complex projective space to another are constructed as projections of the Segre map of the complexifi- cation. A classification theorem relates equivalence classes of projections to congru- ence classes of matrix pencils. Maps from the 2-sphere to the complex projective plane, which generalize stereographic projection, and immersions of the complex projective plane in four and five complex dimensions, are considered in detail. Of particular interest are the CR singular points in the image. MSC 2000: 14E05, 14P05, 15A22, 32S20, 32V40 1. Introduction It was shown by [23] that the complex projective plane CP 2 can be embedded in R7. An example of such an embedding, where R7 is considered as a subspace of C4, and CP 2 has complex homogeneous coordinates [z1 : z2 : z3], was given by the following parametric map: 1 2 2 [z1 : z2 : z3] 7→ 2 2 2 (z2z¯3, z3z¯1, z1z¯2, |z1| − |z2| ). |z1| + |z2| + |z3| Another parametric map of a similar form embeds the complex projective line CP 1 in R3 ⊆ C2: 1 2 2 [z0 : z1] 7→ 2 2 (2¯z0z1, |z1| − |z0| ). |z0| + |z1| ∗The author’s research was supported in part by a 1999 IPFW Summer Faculty Research Grant. 0138-4821/93 $ 2.50 c 2002 Heldermann Verlag 452 Adam Coffman: CR Singular Immersions of Complex Projective Spaces This may look more familiar when restricted to an affine neighborhood, [z0 : z1] = (1, z) = (1, x + iy), so the set of complex numbers is mapped to the unit sphere: 2x 2y |z|2 − 1 z 7→ ( , , ), 1 + |z|2 1 + |z|2 1 + |z|2 and the “point at infinity”, [0 : 1], is mapped to the point (0, 0, 1) ∈ R3. -
Equivariant Cohomology Chern Numbers Determine Equivariant
EQUIVARIANT COHOMOLOGY CHERN NUMBERS DETERMINE EQUIVARIANT UNITARY BORDISM FOR TORUS GROUPS ZHI LÜ AND WEI WANG ABSTRACT. This paper shows that the integral equivariant cohomology Chern num- bers completely determine the equivariant geometric unitary bordism classes of closed unitary G-manifolds, which gives an affirmative answer to the conjecture posed by Guillemin–Ginzburg–Karshon in [20, Remark H.5, §3, Appendix H], where G is a torus. As a further application, we also obtain a satisfactory solution of [20, Question (A), §1.1, Appendix H] on unitary Hamiltonian G-manifolds. Our key ingredients in the proof are the universal toric genus defined by Buchstaber–Panov–Ray and the Kronecker pairing of bordism and cobordism. Our approach heavily exploits Quillen’s geometric inter- pretation of homotopic unitary cobordism theory. Moreover, this method can also be k applied to the study of (Z2) -equivariant unoriented bordism and can still derive the classical result of tom Dieck. 1. INTRODUCTION AND MAIN RESULTS 1.1. Background. In his seminal work [37] , R. Thom introduced the unoriented bor- dism theory, which corresponds to the infinite orthogonal group O. Since then, vari- ous other bordism theories, which correspond to subgroups G of the orthogonal group O as structure groups of stable tangent bundles or stable normal bundles of compact smooth manifolds, have been studied and established (e.g., see [41, 31, 32] and for more details, see [26, 35]). When G is chosen as SO (resp. U, SU etc.), the corresponding bor- dism theory is often called the oriented (resp. unitary, special unitary etc.) bordism theory.