String Structures Associated to Indefinite Lie Groups
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The Stable Homotopy of Complex Projective Space
THE STABLE HOMOTOPY OF COMPLEX PROJECTIVE SPACE By GRAEME SEGAL [Received 17 March 1972] 1. Introduction THE object of this note is to prove that the space BU is a direct factor of the space Q(CP°°) = Oto5eo(CP00) = HmQn/Sn(CPto). This is not very n surprising, as Toda [cf. (6) (2.1)] has shown that the homotopy groups of ^(CP00), i.e. the stable homotopy groups of CP00, split in the appro- priate way. But the method, which is Quillen's technique (7) of reducing to a problem about finite groups and then using the Brauer induction theorem, may be interesting. If X and Y are spaces, I shall write {X;7}*for where Y+ means Y together with a disjoint base-point, and [ ; ] means homotopy classes of maps with no conditions about base-points. For fixed Y, X i-> {X; Y}* is a representable cohomology theory. If Y is a topological abelian group the composition YxY ->Y induces and makes { ; Y}* into a multiplicative cohomology theory. In fact it is easy to see that {X; Y}° is then even a A-ring. Let P = CP00, and embed it in the space ZxBU which represents the functor K by P — IXBQCZXBU. This corresponds to the natural inclusion {line bundles} c {virtual vector bundles}. There is an induced map from the suspension-spectrum of P to the spectrum representing l£-theory, inducing a transformation of multiplicative cohomology theories T: { ; P}* -*• K*. PROPOSITION 1. For any space, X the ring-homomorphism T:{X;P}°-*K<>(X) is mirjective. COEOLLAKY. The space QP is (up to homotopy) the product of BU and a space with finite homotopy groups. -
Some Stuff About the String Gerbe
Some Stuff about the String Gerbe Danny Stevenson Department of Mathematics, University of California, Riverside Riverside, CA 92521, USA email: [email protected] March 14, 2006 1 Introduction Let G be a compact, simple and simply connected Lie group and let π : P → M be a principal G bundle over a smooth manifold M. Let ν denote the universally transgressive generator of H3(G; Z) = Z and let c ∈ H4(BG; Z) = H3(G; Z) be the transgression of ν. By regarding H4(BG; Z) as a lattice in (S2g∗)G we will allow ourselves to confuse c and the basic inner product h , i on g. Recall [23] that h ,√i is the Killing form on g, normalised so that the longest root θ has length 2. In the physics literature, M is said to be string, or admit a string structure, if a certain characteristic class in H3(LM; Z) vanishes (here LM denotes the free loop space of M). This characteristic class is the obstruction to lifting the structure group of the principal LG- bundle LP → LM to LGc — the Kac-Moody group. As has been observed by several authors [12, 13, 19] the obstruction in H3(LM; Z) is closely related to the characteristic class c ∈ H4(M; Z): if M is 2-connected a lift of the structure group to LGc exists precisely when the map c: M → K(Z, 4) is null- homotopic. As is well known, if G = Spin(n) then 2c = p1. This obstruction problem on LM can be phrased in the language of homotopy theory down on M. -
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 Construction of String 2-Group Modelsusing a Transgression
A Construction of String 2-Group Models using a Transgression-Regression Technique Konrad Waldorf Fakult¨at f¨ur Mathematik Universit¨at Regensburg Universit¨atsstraße 31 D–93053 Regensburg [email protected] Dedicated to Steven Rosenberg on the occasion of his 60th birthday Abstract In this note we present a new construction of the string group that ends optionally in two different contexts: strict diffeological 2-groups or finite-dimensional Lie 2-groups. It is canonical in the sense that no choices are involved; all the data is written down and can be looked up (at least somewhere). The basis of our construction is the basic gerbe of Gaw¸edzki-Reis and Meinrenken. The main new insight is that under a transgression- regression procedure, the basic gerbe picks up a multiplicative structure coming from the Mickelsson product over the loop group. The conclusion of the construction is a relation between multiplicative gerbes and 2-group extensions for which we use recent work of Schommer-Pries. MSC 2010: Primary 22E67; secondary 53C08, 81T30, 58H05 arXiv:1201.5052v2 [math.DG] 4 May 2012 1 Introduction The string group String(n) is a topological group defined up to homotopy equivalence as the 3-connected cover of Spin(n), for n =3 or n> 4. Concrete models for String(n) have been provided by Stolz [Sto96] and Stolz-Teichner [ST04]. In order to understand, e.g. the differential geometry of String(n), the so-called “string geometry”, it is necessary to have models in better categories than topological groups. Its 3-connectedness implies that String(n) is a K(Z, 2)-fibration over Spin(n), so that it cannot be a (finite-dimensional) Lie group. -
Notes on Principal Bundles and Classifying Spaces
Notes on principal bundles and classifying spaces Stephen A. Mitchell August 2001 1 Introduction Consider a real n-plane bundle ξ with Euclidean metric. Associated to ξ are a number of auxiliary bundles: disc bundle, sphere bundle, projective bundle, k-frame bundle, etc. Here “bundle” simply means a local product with the indicated fibre. In each case one can show, by easy but repetitive arguments, that the projection map in question is indeed a local product; furthermore, the transition functions are always linear in the sense that they are induced in an obvious way from the linear transition functions of ξ. It turns out that all of this data can be subsumed in a single object: the “principal O(n)-bundle” Pξ, which is just the bundle of orthonormal n-frames. The fact that the transition functions of the various associated bundles are linear can then be formalized in the notion “fibre bundle with structure group O(n)”. If we do not want to consider a Euclidean metric, there is an analogous notion of principal GLnR-bundle; this is the bundle of linearly independent n-frames. More generally, if G is any topological group, a principal G-bundle is a locally trivial free G-space with orbit space B (see below for the precise definition). For example, if G is discrete then a principal G-bundle with connected total space is the same thing as a regular covering map with G as group of deck transformations. Under mild hypotheses there exists a classifying space BG, such that isomorphism classes of principal G-bundles over X are in natural bijective correspondence with [X, BG]. -
Lectures on the Stable Homotopy of BG 1 Preliminaries
Geometry & Topology Monographs 11 (2007) 289–308 289 arXiv version: fonts, pagination and layout may vary from GTM published version Lectures on the stable homotopy of BG STEWART PRIDDY This paper is a survey of the stable homotopy theory of BG for G a finite group. It is based on a series of lectures given at the Summer School associated with the Topology Conference at the Vietnam National University, Hanoi, August 2004. 55P42; 55R35, 20C20 Let G be a finite group. Our goal is to study the stable homotopy of the classifying space BG completed at some prime p. For ease of notation, we shall always assume that any space in question has been p–completed. Our fundamental approach is to decompose the stable type of BG into its various summands. This is useful in addressing many questions in homotopy theory especially when the summands can be identified with simpler or at least better known spaces or spectra. It turns out that the summands of BG appear at various levels related to the subgroup lattice of G. Moreover since we are working at a prime, the modular representation theory of automorphism groups of p–subgroups of G plays a key role. These automorphisms arise from the normalizers of these subgroups exactly as they do in p–local group theory. The end result is that a complete stable decomposition of BG into indecomposable summands can be described (Theorem 6) and its stable homotopy type can be characterized algebraically (Theorem 7) in terms of simple modules of automorphism groups. This paper is a slightly expanded version of lectures given at the International School of the Hanoi Conference on Algebraic Topology, August 2004. -
The Classifying Space of the G-Cobordism Category in Dimension Two
The classifying space of the G-cobordism category in dimension two Carlos Segovia Gonz´alez Instituto de Matem´aticasUNAM Oaxaca, M´exico 29 January 2019 The classifying space I Graeme Segal, Classifying spaces and spectral sequences (1968) I A. Grothendieck, Th´eorie de la descente, etc., S´eminaire Bourbaki,195 (1959-1960) I S. Eilenberg and S. MacLane, Relations between homology and homotopy groups of spaces I and II (1945-1950) I P.S. Aleksandrov-E.Chec,ˇ Nerve of a covering (1920s) I B. Riemann, Moduli space (1857) Definition A simplicial set X is a contravariant functor X : ∆ ! Set from the simplicial category to the category of sets. The geometric realization, denoted by jX j, is the topological space defined as ` jX j := n≥0(∆n × Xn)= ∼, where (s; X (f )a) ∼ (∆f (s); a) for all s 2 ∆n, a 2 Xm, and f :[n] ! [m] in ∆. Definition For a small category C we define the simplicial set N(C), called the nerve, denoted by N(C)n := Fun([n]; C), which consists of all the functors from the category [n] to C. The classifying space of C is the geometric realization of N(C). Denote this by BC := jN(C)j. Properties I The functor B : Cat ! Top sends a category to a topological space, a functor to a continuous map and a natural transformation to a homotopy. 0 0 I B conmutes with products B(C × C ) =∼ BC × BC . I A equivalence of categories gives a homotopy equivalence. I A category with initial or final object is contractil. -
Spaces of Graphs and Surfaces: on the Work of Søren Galatius
BULLETIN (New Series) OF THE AMERICAN MATHEMATICAL SOCIETY Volume 49, Number 1, January 2012, Pages 73–90 S 0273-0979(2011)01360-X Article electronically published on October 19, 2011 SPACES OF GRAPHS AND SURFACES: ON THE WORK OF SØREN GALATIUS ULRIKE TILLMANN Abstract. We put Soren Galatius’s result on the homology of the automor- phism group of free groups into context. In particular we explain its relation to the Mumford conjecture and the main ideas of the proofs. 1. Introduction Galatius’s most striking result is easy enough to state. Let Σn be the symmetric group on n letters, and let Fn be the free (non-abelian) group with n generators. The symmetric group Σn acts naturally by permutation on the n generators of Fn, and every permutation thus gives rise to an automorphism of the free group. Galatius proves that the map Σn → AutFn in homology induces an isomorphism in degrees less than (n − 1)/2 and in rational homology in degrees less than 2n/3. The homology of the symmetric groups in these ranges is well understood. In particular, in common with all finite groups, it has no non-trivial rational homology. By Galatius’s theorem, in low degrees this is then also true for AutFn: H∗(AutFn) ⊗ Q =0 for 0< ∗ < 2n/3, as had been conjectured by Hatcher and Vogtmann. We will put this result in context and explain the connection with previous work on the mapping class group of surfaces and the homotopy theoretic approach to a conjecture by Mumford in [37] on its rational, stable cohomology. -
Classifying Spaces and Spectral Sequences Graeme Segal
CLASSIFYING SPACES AND SPECTRAL SEQUENCES GRAEME SEGAL The following work makes no great claim to originality. The first three sections are devoted to a very general discussion of the representation of categories by topological spaces, and all the ideas are implicit in the work of Grothendieck. But I think the essential simplicity of the situation has never been made quite explicit, and I hope the present popularization will be of some interest. Apart from this my purpose is to obtain for a generalized cohomology theory k* a spectral sequence connecting A*(X) with the ordinary cohomology of X. This has been done in the past [i], when X is a GW-complex, by considering the filtration ofX by its skeletons. I give a construction which makes no such assumption on X: the interest of this is that it works also in the case of an equivariant cohomology theory defined on a category of G-spaces, where G is a fixed topological group. But I have not discussed that application here, and I refer the reader to [13]. On the other hand I have explained in detail the context into which the construction fits, and its relation to other spectral sequences obtained in [8] and [12] connected with the bar-construction. § i. SEMI-SIMPLICIAL OBJECTS A semi-simplicial set is a sequence of sets AQ, A^, Ag, ... together with boundary - and degeneracy-maps which satisfy certain well-known conditions [5]. But it is better regarded as a contravariant functor A from the category Ord of finite totally ordered sets to the category of sets. -
Configuration-Spaces and Iterated Loop-Spaces
Inventiones math. 21,213- 221 (1973) by Springer-Verlag 1973 Configuration-Spaces and Iterated Loop-Spaces Graeme Segal (Oxford) w 1. Introduction The object of this paper is to prove a theorem relating "configuration- spaces" to iterated loop-spaces. The idea of the connection between them seems to be due to Boardman and Vogt [2]. Part of the theorem has been proved by May [6]; the general case has been announced by Giffen [4], whose method is to deduce it from the work of Milgram [7]. Let C. be the space of finite subsets of ~n. It is topologized as the disjoint union LI C~,k, where C~, k is the space of subsets of cardinal k, k->0 regarded as the orbit-space of the action of the symmetric group 2:k on the space ~n,k of ordered subsets of cardinal k, which is an open subset of R "k. There is a map from C. to O"S ~, the space of base-point preserving maps S"--,S ~, where S" is the n-sphere. One description of it (at least when n> 1) is as follows. Think of a finite subset c of ~ as a set of electrically charged particles, each of charge + 1, and associate to it the electric field E c it generates. This is a map Ec: R~- c-.n~ ~ which can be extended to a continuous map Ec: RnUOO--,~nU~ by defining Ec(~)=oo if ~ec, and Ec(oo)=0. Then E c can be regarded as a base-point-preserving map S"--,S", where the base-point is oo on the left and 0 on the right. -
On the String 2-Group
On the String 2-Group Urs Schreiber∗ May 3, 2006 Oberseminar Topologie Universit¨atBonn, May 2, 2006 Abstract We try to convey the main idea for 1. • what the String-group StringG is • and how it is the nerve of a 2-group StrG; 2. as well as • what a StrG-2-bundle is • and how it is “the same” as a StringG-bundle. The first point is due to [10, 9], which will be reviewed in section 2. The second point has been addressed in [8] using the language of bundle gerbes. In section 3 we review this, using a 2-functorial language which is natural with respect to the 2-group nature of StrG. A commented bibliography is given in section 1. Contents 1 Motivation 2 2 The String 2-Group 3 2.1 2-Groups . 4 2.2 StringG ................................ 6 3 String 2-Bundles 9 3.1 2-Bundles . 9 3.2 Transition Bundles . 15 4 Outlook: String Connections 19 ∗e-mail: urs.schreiber at math.uni-hamburg.de 1 1 Motivation The main motivation for the following discussion has its origin in theoretical physics. Elementary particles with spin are described by sections of spin bundles. From the physical point of view, the necessity of a spin structure on spacetime may be deduced from a certain global anomaly for the path integral of a single, point-like, fermion. The path integral (albeit a somewhat heuristic device) can be regarded as a single valued function on the space of configurations of the particle, only if the (first and) second Stiefel-Whitney class of spacetime vanishes. -
Classifying Foliations
Classifying foliations Steven Hurder Abstract. We give a survey of the approaches to classifying foliations, start- ing with the Haefliger classifying spaces and the various results and examples about the secondary classes of foliations. Various dynamical properties of fo- liations are introduced and discussed, including expansion rate, local entropy, and orbit growth rates. This leads to a decomposition of the foliated space into Borel or measurable components with these various dynamical types. The dynamical structure is compared with the classification via secondary classes. 1991 Mathematics Subject Classification. Primary 22F05, 37C85, 57R20, 57R32, 58H05, 58H10; Secondary 37A35, 37A55, 37C35 . Key words and phrases. Foliations, differentiable groupoids, smooth dynamical systems, er- godic theory, classifying spaces, secondary characteristic classes . The author was supported in part by NSF Grant #0406254. 1 2 STEVEN HURDER 1. Introduction A basic problem of foliation theory is how to \classify all the foliations" of fixed codimension-q on a given closed manifold M, assuming that at least one such foliation exists on M. This survey concerns this classification problem for foliations. Kaplan [192] proved the first complete classification result in the subject in 1941. For a foliation F of the plane by lines (no closed orbits) the leaf space T = R2=F is a (possibly non-Hausdorff) 1-manifold, and F is characterized up to homeomorphism by the leaf space T . (See also [126, 127, 324].) Palmeira [252] proved an analogous result for the case of foliations of simply connected manifolds by hyperplanes. Research on classification advanced dramatically in 1970, with three seminal works: Bott's Vanishing Theorem [28], Haefliger’s construction of a \classifying space" for foliations [119, 120], and Thurston's profound results on existence and classifi- cation of foliations [297, 298, 299, 300], in terms of the homotopy theory of Haefliger’s classifying spaces BΓq.