Kreck-Stolz Invariants for Quaternionic Line Bundles
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On the Stable Classification of Certain 4-Manifolds
BULL. AUSTRAL. MATH. SOC. 57N65, 57R67, 57Q10, 57Q20 VOL. 52 (1995) [385-398] ON THE STABLE CLASSIFICATION OF CERTAIN 4-MANIFOLDS ALBERTO CAVICCHIOLI, FRIEDRICH HEGENBARTH AND DUSAN REPOVS We study the s-cobordism type of closed orientable (smooth or PL) 4—manifolds with free or surface fundamental groups. We prove stable classification theorems for these classes of manifolds by using surgery theory. 1. INTRODUCTION In this paper we shall study closed connected (smooth or PL) 4—manifolds with special fundamental groups as free products or surface groups. For convenience, all manifolds considered will be assumed to be orientable although our results work also in the general case, provided the first Stiefel-Whitney classes coincide. The starting point for classifying manifolds is the determination of their homotopy type. For 4- manifolds having finite fundamental groups with periodic homology of period four, this was done in [12] (see also [1] and [2]). The case of a cyclic fundamental group of prime order was first treated in [23]. The homotopy type of 4-manifolds with free or surface fundamental groups was completely classified in [6] and [7] respectively. In particular, closed 4-manifolds M with a free fundamental group IIi(M) = *PZ (free product of p factors Z ) are classified, up to homotopy, by the isomorphism class of their intersection pairings AM : B2{M\K) x H2(M;A) —> A over the integral group ring A = Z[IIi(M)]. For Ui(M) = Z, we observe that the arguments developed in [11] classify these 4-manifolds, up to TOP homeomorphism, in terms of their intersection forms over Z. -
Notes on the Atiyah-Singer Index Theorem Liviu I. Nicolaescu
Notes on the Atiyah-Singer Index Theorem Liviu I. Nicolaescu Notes for a topics in topology course, University of Notre Dame, Spring 2004, Spring 2013. Last revision: November 15, 2013 i The Atiyah-Singer Index Theorem This is arguably one of the deepest and most beautiful results in modern geometry, and in my view is a must know for any geometer/topologist. It has to do with elliptic partial differential opera- tors on a compact manifold, namely those operators P with the property that dim ker P; dim coker P < 1. In general these integers are very difficult to compute without some very precise information about P . Remarkably, their difference, called the index of P , is a “soft” quantity in the sense that its determination can be carried out relying only on topological tools. You should compare this with the following elementary situation. m n Suppose we are given a linear operator A : C ! C . From this information alone we cannot compute the dimension of its kernel or of its cokernel. We can however compute their difference which, according to the rank-nullity theorem for n×m matrices must be dim ker A−dim coker A = m − n. Michael Atiyah and Isadore Singer have shown in the 1960s that the index of an elliptic operator is determined by certain cohomology classes on the background manifold. These cohomology classes are in turn topological invariants of the vector bundles on which the differential operator acts and the homotopy class of the principal symbol of the operator. Moreover, they proved that in order to understand the index problem for an arbitrary elliptic operator it suffices to understand the index problem for a very special class of first order elliptic operators, namely the Dirac type elliptic operators. -
Finite Group Actions on Kervaire Manifolds 3
FINITE GROUP ACTIONS ON KERVAIRE MANIFOLDS DIARMUID CROWLEY AND IAN HAMBLETON M4k+2 Abstract. Let K be the Kervaire manifold: a closed, piecewise linear (PL) mani- fold with Kervaire invariant 1 and the same homology as the product S2k+1 × S2k+1 of M4k+2 spheres. We show that a finite group of odd order acts freely on K if and only if 2k+1 2k+1 it acts freely on S × S . If MK is smoothable, then each smooth structure on M j M4k+2 K admits a free smooth involution. If k 6= 2 − 1, then K does not admit any M30 M62 free TOP involutions. Free “exotic” (PL) involutions are constructed on K , K , and M126 M30 Z Z K . Each smooth structure on K admits a free /2 × /2 action. 1. Introduction One of the main themes in geometric topology is the study of smooth manifolds and their piece-wise linear (PL) triangulations. Shortly after Milnor’s discovery [54] of exotic smooth 7-spheres, Kervaire [39] constructed the first example (in dimension 10) of a PL- manifold with no differentiable structure, and a new exotic smooth 9-sphere Σ9. The construction of Kervaire’s 10-dimensional manifold was generalized to all dimen- sions of the form m ≡ 2 (mod 4), via “plumbing” (see [36, §8]). Let P 4k+2 denote the smooth, parallelizable manifold of dimension 4k+2, k ≥ 0, constructed by plumbing two copies of the the unit tangent disc bundle of S2k+1. The boundary Σ4k+1 = ∂P 4k+2 is a smooth homotopy sphere, now usually called the Kervaire sphere. -
Mapping Surgery to Analysis III: Exact Sequences
K-Theory (2004) 33:325–346 © Springer 2005 DOI 10.1007/s10977-005-1554-7 Mapping Surgery to Analysis III: Exact Sequences NIGEL HIGSON and JOHN ROE Department of Mathematics, Penn State University, University Park, Pennsylvania 16802. e-mail: [email protected]; [email protected] (Received: February 2004) Abstract. Using the constructions of the preceding two papers, we construct a natural transformation (after inverting 2) from the Browder–Novikov–Sullivan–Wall surgery exact sequence of a compact manifold to a certain exact sequence of C∗-algebra K-theory groups. Mathematics Subject Classifications (1991): 19J25, 19K99. Key words: C∗-algebras, L-theory, Poincare´ duality, signature operator. This is the final paper in a series of three whose objective is to construct a natural transformation from the surgery exact sequence of Browder, Novikov, Sullivan and Wall [17,21] to a long exact sequence of K-theory groups associated to a certain C∗-algebra extension; we finally achieve this objective in Theorem 5.4. In the first paper [5], we have shown how to associate a homotopy invariant C∗-algebraic signature to suitable chain complexes of Hilbert modules satisfying Poincare´ duality. In the second paper, we have shown that such Hilbert–Poincare´ complexes arise natu- rally from geometric examples of manifolds and Poincare´ complexes. The C∗-algebras that are involved in these calculations are analytic reflections of the equivariant and/or controlled structure of the underlying topology. In paper II [6] we have also clarified the relationship between the analytic signature, defined by the procedure of paper I for suitable Poincare´ com- plexes, and the analytic index of the signature operator, defined only for manifolds. -
FOLIATIONS Introduction. the Study of Foliations on Manifolds Has a Long
BULLETIN OF THE AMERICAN MATHEMATICAL SOCIETY Volume 80, Number 3, May 1974 FOLIATIONS BY H. BLAINE LAWSON, JR.1 TABLE OF CONTENTS 1. Definitions and general examples. 2. Foliations of dimension-one. 3. Higher dimensional foliations; integrability criteria. 4. Foliations of codimension-one; existence theorems. 5. Notions of equivalence; foliated cobordism groups. 6. The general theory; classifying spaces and characteristic classes for foliations. 7. Results on open manifolds; the classification theory of Gromov-Haefliger-Phillips. 8. Results on closed manifolds; questions of compact leaves and stability. Introduction. The study of foliations on manifolds has a long history in mathematics, even though it did not emerge as a distinct field until the appearance in the 1940's of the work of Ehresmann and Reeb. Since that time, the subject has enjoyed a rapid development, and, at the moment, it is the focus of a great deal of research activity. The purpose of this article is to provide an introduction to the subject and present a picture of the field as it is currently evolving. The treatment will by no means be exhaustive. My original objective was merely to summarize some recent developments in the specialized study of codimension-one foliations on compact manifolds. However, somewhere in the writing I succumbed to the temptation to continue on to interesting, related topics. The end product is essentially a general survey of new results in the field with, of course, the customary bias for areas of personal interest to the author. Since such articles are not written for the specialist, I have spent some time in introducing and motivating the subject. -
A Combinatorial Computation of the First Pontryagin Class of the Complex Projective Plane
LAZAR MILIN A COMBINATORIAL COMPUTATION OF THE FIRST PONTRYAGIN CLASS OF THE COMPLEX PROJECTIVE PLANE ABSTRACT. This paper carries out an explicit computation of the combinatorial formula of Gabrielov, Gel'fand, and Losik for the first Pontryagin class of the complex projective plane with the 9-vertex triangulation discovered by Wolfgang Kfihnel. The conditions of the original formula must be modified since the 8-vertex triangulation of the 3-sphere link of each vertex cannot be realized as the complex of faces of a convex polytope in 4-space, but it can be so realized by a star-shaped polytope, and the space of all such realizations is not connected. 0. INTRODUCTION In a series of papers [24]-[26] in the 1940s, L. S. Pontryagin defined and studied 'characteristic cycles' on smooth manifolds. In the introduction to the first paper of the series, he stressed the importance of having 'a definition of characteristic cycles which would be applicable to combinatorial mani- folds' since it would provide an algorithm for their calculation from 'the combinatorial structure of the manifolds'. In the late 1950s, Rdne Thom [33], and independently Rohlin and Sarc [28], proved that the rational Pontryagin classes are combinatorial invariants, and in the 1960s. S. P. Novikov [23] proved that these classes are topological invariants. No combinatorial formula for Pontryagin classes was proved until the mid-1970s, when Gabrielov et al. [6], [7] established a formula for the first Pontryagin class pl(X) of a combinatorial manifold X. The formula ex- presses Pl(X) in terms of the simplicial structure of X and some additional structure imposed on X, so in this sense the formula is not purely combinatorial. -
Vector Bundles. Characteristic Classes. Cobordism. Applications
Math 754 Chapter IV: Vector Bundles. Characteristic classes. Cobordism. Applications Laurenţiu Maxim Department of Mathematics University of Wisconsin [email protected] May 3, 2018 Contents 1 Chern classes of complex vector bundles 2 2 Chern classes of complex vector bundles 2 3 Stiefel-Whitney classes of real vector bundles 5 4 Stiefel-Whitney classes of manifolds and applications 5 4.1 The embedding problem . .6 4.2 Boundary Problem. .9 5 Pontrjagin classes 11 5.1 Applications to the embedding problem . 14 6 Oriented cobordism and Pontrjagin numbers 15 7 Signature as an oriented cobordism invariant 17 8 Exotic 7-spheres 19 9 Exercises 20 1 1 Chern classes of complex vector bundles 2 Chern classes of complex vector bundles We begin with the following Proposition 2.1. ∗ ∼ H (BU(n); Z) = Z [c1; ··· ; cn] ; with deg ci = 2i ∗ Proof. Recall that H (U(n); Z) is a free Z-algebra on odd degree generators x1; ··· ; x2n−1, with deg(xi) = i, i.e., ∗ ∼ H (U(n); Z) = ΛZ[x1; ··· ; x2n−1]: Then using the Leray-Serre spectral sequence for the universal U(n)-bundle, and using the fact that EU(n) is contractible, yields the desired result. Alternatively, the functoriality of the universal bundle construction yields that for any subgroup H < G of a topological group G, there is a fibration G=H ,! BH ! BG. In our A 0 case, consider U(n − 1) as a subgroup of U(n) via the identification A 7! . Hence, 0 1 there exists fibration U(n)=U(n − 1) ∼= S2n−1 ,! BU(n − 1) ! BU(n): Then the Leray-Serre spectral sequence and induction on n gives the desired result, where 1 ∗ 1 ∼ we use the fact that BU(1) ' CP and H (CP ; Z) = Z[c] with deg c = 2. -
Surgery on Compact Manifolds, Second Edition, 1999 68 David A
http://dx.doi.org/10.1090/surv/069 Selected Titles in This Series 69 C. T. C. Wall (A. A. Ranicki, Editor), Surgery on compact manifolds, Second Edition, 1999 68 David A. Cox and Sheldon Katz, Mirror symmetry and algebraic geometry, 1999 67 A. Borel and N. Wallach, Continuous cohomology, discrete subgroups, and representations of reductive groups, Second Edition, 1999 66 Yu. Ilyashenko and Weigu Li, Nonlocal bifurcations, 1999 65 Carl Faith, Rings and things and a fine array of twentieth century associative algebra, 1999 64 Rene A. Carmona and Boris Rozovskii, Editors, Stochastic partial differential equations: Six perspectives, 1999 63 Mark Hovey, Model categories, 1999 62 Vladimir I. Bogachev, Gaussian measures, 1998 61 W. Norrie Everitt and Lawrence Markus, Boundary value problems and symplectic algebra for ordinary differential and quasi-differential operators, 1999 60 Iain Raeburn and Dana P. Williams, Morita equivalence and continuous-trace C*-algebras, 1998 59 Paul Howard and Jean E. Rubin, Consequences of the axiom of choice, 1998 58 Pavel I. Etingof, Igor B. Frenkel, and Alexander A. Kirillov, Jr., Lectures on representation theory and Knizhnik-Zamolodchikov equations, 1998 57 Marc Levine, Mixed motives, 1998 56 Leonid I. Korogodski and Yan S. Soibelman, Algebras of functions on quantum groups: Part I, 1998 55 J. Scott Carter and Masahico Saito, Knotted surfaces and their diagrams, 1998 54 Casper Goffman, Togo Nishiura, and Daniel Waterman, Homeomorphisms in analysis, 1997 53 Andreas Kriegl and Peter W. Michor, The convenient setting of global analysis, 1997 52 V. A. Kozlov, V. G. Maz'ya> and J. Rossmann, Elliptic boundary value problems in domains with point singularities, 1997 51 Jan Maly and William P. -
Milnor, John W. Groups of Homotopy Spheres
BULLETIN (New Series) OF THE AMERICAN MATHEMATICAL SOCIETY Volume 52, Number 4, October 2015, Pages 699–710 http://dx.doi.org/10.1090/bull/1506 Article electronically published on June 12, 2015 SELECTED MATHEMATICAL REVIEWS related to the paper in the previous section by JOHN MILNOR MR0148075 (26 #5584) 57.10 Kervaire, Michel A.; Milnor, John W. Groups of homotopy spheres. I. Annals of Mathematics. Second Series 77 (1963), 504–537. The authors aim to study the set of h-cobordism classes of smooth homotopy n-spheres; they call this set Θn. They remark that for n =3 , 4thesetΘn can also be described as the set of diffeomorphism classes of differentiable structures on Sn; but this observation rests on the “higher-dimensional Poincar´e conjecture” plus work of Smale [Amer. J. Math. 84 (1962), 387–399], and it does not really form part of the logical structure of the paper. The authors show (Theorem 1.1) that Θn is an abelian group under the connected sum operation. (In § 2, the authors give a careful treatment of the connected sum and of the lemmas necessary to prove Theorem 1.1.) The main task of the present paper, Part I, is to set up methods for use in Part II, and to prove that for n = 3 the group Θn is finite (Theorem 1.2). (For n =3the authors’ methods break down; but the Poincar´e conjecture for n =3wouldimply that Θ3 = 0.) We are promised more detailed information about the groups Θn in Part II. The authors’ method depends on introducing a subgroup bPn+1 ⊂ Θn;asmooth homotopy n-sphere qualifies for bPn+1 if it is the boundary of a parallelizable man- ifold. -
Higher Anomalies, Higher Symmetries, and Cobordisms I
Higher Anomalies, Higher Symmetries, and Cobordisms I: Classification of Higher-Symmetry-Protected Topological States and Their Boundary Fermionic/Bosonic Anomalies via a Generalized Cobordism Theory Zheyan Wan1;2 and Juven Wang3;4 1 1Yau Mathematical Sciences Center, Tsinghua University, Beijing 100084, China 2School of Mathematical Sciences, USTC, Hefei 230026, China 3School of Natural Sciences, Institute for Advanced Study, Einstein Drive, Princeton, NJ 08540, USA 4Center of Mathematical Sciences and Applications, Harvard University, Cambridge, MA 02138, USA Abstract By developing a generalized cobordism theory, we explore the higher global symmetries and higher anomalies of quantum field theories and interacting fermionic/bosonic systems in condensed matter. Our essential math input is a generalization of Thom-Madsen-Tillmann spectra, Adams spectral sequence, and Freed-Hopkins's theorem, to incorporate higher-groups and higher classifying spaces. We provide many examples of bordism groups with a generic H-structure manifold with a higher-group G, and their bordism invariants | e.g. perturbative anomalies of chiral fermions [originated from Adler-Bell-Jackiw] or bosons with U(1) symmetry in any even spacetime dimensions; non-perturbative global anomalies such as Witten anomaly and the new SU(2) anomaly in 4d and 5d. Suitable H such as SO/Spin/O/Pin± enables arXiv:1812.11967v2 [hep-th] 11 Jul 2019 the study of quantum vacua of general bosonic or fermionic systems with time-reversal or reflection symmetry on (un)orientable spacetime. Higher 't Hooft anomalies of dd live on the boundary of (d + 1)d higher-Symmetry-Protected Topological states (SPTs) or symmetric invertible topological orders (i.e., invertible topological quantum field theories at low energy); thus our cobordism theory also classifies and characterizes higher-SPTs. -
Arxiv:1506.05408V1 [Math.AT]
NOVIKOV’S CONJECTURE JONATHAN ROSENBERG Abstract. We describe Novikov’s “higher signature conjecture,” which dates back to the late 1960’s, as well as many alternative formulations and related problems. The Novikov Conjecture is perhaps the most important unsolved problem in high-dimensional manifold topology, but more importantly, vari- ants and analogues permeate many other areas of mathematics, from geometry to operator algebras to representation theory. 1. Origins of the Original Conjecture The Novikov Conjecture is perhaps the most important unsolved problem in the topology of high-dimensional manifolds. It was first stated by Sergei Novikov, in various forms, in his lectures at the International Congresses of Mathematicians in Moscow in 1966 and in Nice in 1970, and in a few other papers [84, 87, 86, 85]. For an annotated version of the original formulation, in both Russian and English, we refer the reader to [37]. Here we will try instead to put the problem in context and explain why it might be of interest to the average mathematician. For a nice book- length exposition of this subject, we recommend [65]. Many treatments of various aspects of the problem can also be found in the many papers in the collections [38, 39]. For the typical mathematician, the most important topological spaces are smooth manifolds, which were introduced by Riemann in the 1850’s. However, it took about 100 years for the tools for classifying manifolds (except in dimension 1, which is trivial, and dimension 2, which is relatively easy) to be developed. The problem is that manifolds have no local invariants (except for the dimension); all manifolds of the same dimension look the same locally. -
On the Construction and Topological Invariance of the Pontryagin Classes
ON THE CONSTRUCTION AND TOPOLOGICAL INVARIANCE OF THE PONTRYAGIN CLASSES ANDREW RANICKI AND MICHAEL WEISS Abstract. We use sheaves and algebraic L-theory to construct the rational Pontryagin classes of fiber bundles with fiber Rn. This amounts to an alternative proof of Novikov’s theorem on the topological invariance of the rational Pontryagin classes of vector bundles. Transversality arguments and torus tricks are avoided. 1. Introduction The “topological invariance of the rational Pontryagin classes” was originally the state- ment that for a homeomorphism of smooth manifolds, f : N → N ′, the induced map f ∗ : H∗(N ′; Q) → H∗(N; Q) ′ 4i ′ ′ takes the Pontryagin classes pi(TN ) ∈ H (N ; Q) of the tangent bundle of N to the Pontryagin classes pi(TN) of the tangent bundle of N. The topological invariance was proved by Novikov [Nov], some 40 years ago. This breakthrough result and its torus-related method of proof have stimulated many subsequent developments in topological manifolds, notably the formulation of the Novikov conjecture, the Kirby-Siebenmann structure theory and the Chapman-Ferry-Quinn et al. controlled topology. The topological invariance of the rational Pontrjagin classes has been reproved several times (Gromov [Gr], Ranicki [Ra3], Ranicki and Yamasaki [RaYa]). In this paper we give yet another proof, using sheaf-theoretic ideas. We associate to a topological manifold M a kind of “tautological” co-sheaf on M of symmetric Poincar´echain complexes, in such a way that the cobordism class is a topological invariant by construction. We then produce excision and homotopy invariance theorems for the cobordism groups of such co-sheaves, and use the Hirzebruch signature theorem to extract the rational Pontryagin classes of M from the cobordism class of the tautological co-sheaf on M.