The -invariant classifies closed smooth oriented -connected rational homology -spheres up to almost-diffeomorphism, that is, diffeomorphism up to a connected sum with an exotic sphere. It also detects exotic homeomorphisms between such manifolds.
The -invariant also gives information about quaternionic line bundles over a fixed manifold, and we use it to give a new proof of a theorem of Feder and Gitler about the values of the second Chern classes of quaternionic line bundles over.
The -invariant for is closely related to the Adams -invariant on the -stem. References [Enhancements On Off] What's this? IVTopology 521— AndersonE. Brown Jr. PetersonThe structure of the Spin cobordism ringAnn. Anderson, W. Benjamin, Inc. MR 4. Atiyah and F. HirzebruchRiemann-Roch theorems for differentiable manifoldsBull. AtiyahV. Patodiand I. SingerSpectral asymmetry and Riemannian geometry. IMath. Cambridge Philos. IIMath. IIIMath. Atiyah and I. SingerThe index of elliptic operators on compact manifoldsBull.
MR 8. Ergebnisse der Mathematik und ihrer Grenzgebiete, Band MR Bunke, On the topological contents of eta invariants, preprintarXiv Thesis Ph.
Diarmuid Crowley and Christine M. Owen DearricottA 7-manifold with positive curvatureDuke Math. Harold DonnellySpectral geometry and invariants from differential topologyBull. London Math. James Eells Jr. KuiperAn invariant for certain smooth manifoldsAnn. Pura Appl.
Feder and S. GitlerMappings of quaternionic projective spacesBol.Let G be a simple, simply-connected, compact Lie group and let M be an orientable, simply-connected, closed 4-manifold. A simply-connected 4-manifold is spin if and only if its intersection form is even. In the case of simply-connected 4-manifolds, the spin condition is equivalent to all cup product squares being trivial in mod 2 cohomology. When M is a spin 4-manifold, topologists have been studying the homotopy types of gauge groups over M extensively over the last twenty years.
On the one hand, Theriault showed in [ 16 ] that there is a homotopy equivalence. When M is a non-spin 4-manifold, the author in [ 14 ] showed that there is a homotopy equivalence. A common approach to classifying the homotopy types of gauge groups is as follows. The author would like to thank his supervisor, Professor Stephen Theriault, for his guidance in writing this paper, and thank the Referee for his careful reading and useful comments.
We want to refine this exact sequence so that the last term is replaced by a group. By definition, we have. Then we have. Localize at an odd prime p. A similar argument works for rational localization. There is a fibration. Then there is a diagram. In Sect.
On the other hand, applying the method used in Sect. Since we will frequently refer to the facts established in [ 45 ], it is easier to follow their setting and consider its adjoint.
MathOverflow is a question and answer site for professional mathematicians. It only takes a minute to sign up. With the usual notations, in homotopy we have:. Very little is know about these spaces. And these alone are very delicate problems. For an idea of the complexity of the problem see "Homotopy type of gauge groups of quaternionic line bundles over spheres" by Claudio and Spreafico.
There is one solution, whose application and finer details I will leave you to decide. The same reasoning applies to all such groups. For general results this is about the best that one can do. Sign up to join this community. The best answers are voted up and rise to the top. Home Questions Tags Users Unanswered. Fundamental group of the space of maps into a classifying space Ask Question. Asked 4 years ago.
Active 4 years ago. Viewed times. Bilateral Bilateral 3, 14 14 silver badges 31 31 bronze badges. You can use characteristic classes to study the later space and thus get some information about the fundamental group, see for example arxiv.
Active Oldest Votes. Tyrone Tyrone 2, 1 1 gold badge 12 12 silver badges 24 24 bronze badges. Maybe 1-connected? It's the only work I know of being done studying the gauge groups of principal bundles over spaces of dimension greater than four. Sign up or log in Sign up using Google. Sign up using Facebook. Sign up using Email and Password.
Columbia Symplectic Geometry, Gauge Theory, and Categorification Seminar
We explain this categorification, and briefly motivate it in the context of equivariant cohomology of flag varieties.
The Homotopy Type of Gauge Theoretic Moduli Spaces
Abstract: We examine the moduli problem for real and quaternionic vector bundles over a curve, and we give a gauge-theoretic construction of moduli varieties for such bundles. These moduli varieties are irreducible subsets of real points inside a complex projective variety. We show, moreover, that any two such connected components are homeomorphic.
Abstract: We shall describe some work in progress with John Baldwin concerning Manolescu's unoriented skein exact sequence for knot Floer homology.
Under the right conditions, this sequence can be iterated to give a cube of resolutions that computes HFK. Using twisted coefficients in a Novikov ring greatly simplifies this cube complex, since the homology of any resolution with multiple components vanishes. It is hoped that this approach may yield a new way to compute HFK combinatorially and shed some light its possible relation to Khovanov homology and the Heegaard Floer homology of the double branched cover.
Abstract: I'll recall Beilinson, Lusztig and MacPherson's classical work on the geometric realization of quantum groups of type A by double partial flag varieties. Then I'll present my recent work on the geometric realization of quantum groups of symmetric type by using localized equivariant derived categories of double framed representation varieties associated with a quiver.
Abstract: I'll describe some A-infinity structures associated to Lagrangians and Lagrangian correspondences, particularly A-infinity modules, bimodules, and higher generalizations called n-modules.
The structures can all be described pictorially in terms of quilted strips with markings, which are types of graph associahedra in disguise. The quilted strips are domains for holomorphic quilts a la Wehrheim-Woodwardso they translate into A-infinity structure on the target symplectic manifolds.
So, for example, a sequence of Lagrangian correspondences between symplectic manifolds M and N determines a bimodule of the Fukaya categories of M and N. Abstract: I'll introduce a family of deformations of the Hamiltonian Floer complex on a symplectic manifold which, on passing to homology, recover the "big" quantum homology of the manifold.What is a Manifold? Lesson 12: Fiber Bundles - Formal Description
Using these deformations, one can construct Calabi quasimorphisms on the universal covers of the Hamiltonian diffeomorphism groups of new families of symplectic manifolds, including all one-point blowups. Abstract: We investigate the algebraic structure of knot Floer homology in the context of categorification.
Ozsvath and Szabo gave the first completely algebraic description of knot Floer homology via a cube of resolutions construction. Starting with a braid diagram for a knot, one singularizes or smooths each crossing, then associates an algebra to each resulting singular braid.Keyword Search. The second aim is to apply this result to the p-local higher homotopy commutativity of gauge groups. Although the higher homotopy commutativity of Lie groups in the sense of Williams is already known, the higher homotopy commutativity in the sense of Sugawara is necessary for this application.
The third aim is to resolve the 5-local higher homotopy noncommutativity problem of the exceptional Lie group G 2which has been open for a long time.
In the proof of Proposition 9. But, actually, it is not well-defined. From this failure, the proofs for Proposition 9. The aim of this current article is to prove a weaker version of Theorem 1. Let [Formula presented] be the principal [Formula presented]-bundle over [Formula presented] such that [Formula presented]. The following is a weaker version of Theorem 1.
We denote the largest integer less than or equal to t by [Formula presented]. Theorem 1. Moreover, if [Formula presented], the converse is also true. Proof To show the if part, it is sufficient to show that the wedge sum [Formula presented] extends over the product [Formula presented].
The case when [Formula presented] has already been verified in [4, Section 5]. Suppose [Formula presented]. By Toda's result [3, Section 7], we have homotopy groups of [Formula presented] as follows: [Formula presented] for [Formula presented], where [Formula presented] if [Formula presented]. This implies that, if [Formula presented] and [Formula presented], there is no obstruction to extending a map [Formula presented] over [Formula presented].
It also implies that, for [Formula presented] and a map [Formula presented], the composite [Formula presented] extends over [Formula presented]. Then we obtain the if part by induction and Theorem 1. The proof of the converse in  correctly works for [Formula presented]. Suppose there exists an extension [Formula presented] of [Formula presented], where [Formula presented] and i is the inclusion [Formula presented].
In the rest of this article, we compute the e-invariant  of the obstruction to extending the map f over [Formula presented]. This obstruction is regarded as an element [Formula presented]. The map h factors as the composite of the suspension map [Formula presented] and the inclusion [Formula presented], where [Formula presented] is the homotopy class corresponding to h under the isomorphism [Formula presented].
We denote the restriction of a on [Formula presented] by [Formula presented]. Then the following holds. Then [Formula presented] is an image of the complexification map from the quaternionic K-theory. Then [Formula presented] is an image of the complexification map from the quaternionic K-theory, but [Formula presented] is not. Proof Consider the following commutative diagram induced by the cofiber sequence: [Formula presented] Note that all the groups appearing in this diagram are free abelian.
This implies the vertical maps are injective. As is well-known, the index of the image of the map [Formula presented] is 1 if i is even, and is 2 if i is odd. Now the lemma follows from the above diagram and the fact that the image of [Formula presented] is generated by [Formula presented].Algebraic Topology and Its Applications pp Cite as.
In recent years Gauge theory has been perhaps the most important technique in the study of differentiable structures on four dimensional manifolds. In particular the study of the moduli spaces of anti self dual connections on a principal bundle over a Riemannian four dimensional manifold has yielded the definition, by Donaldson, of polynomial invariants that have been highly successful at distinguishing smooth structures on homeomorphic manifolds.
Thus studying the homotopy type of these moduli spaces and variants of them has been, and continues to be of fundamental importance in Algebraic Topology. Unable to display preview.
Atiyah, Instantons in two and four dimensionsComm. Google Scholar. Press, 1— Atiyah and R. Series A— Atiyah, V. Drinfeld, N. Hitchin and Y. Manin, Construction of instantonsPhys. A 65— Atiyah and N. Hitchin, The geometry and dynamics of magnetic monopolesPrinceton Univ.
Press, Atiyah, J.Show Complete Series. What is the non-perturbative theory formerly known as Strings? Work on formulating the fundamental principles underlying M-theory has noticeably waned. If history is a good guide, then we should expect that anything as profound and far-reaching as a fully satisfactory formulation of M-theory is surely going to lead to new and novel mathematics.
Regrettably, it is a problem the community seems to have put aside — temporarily. Moore, Physical Mathematics and the Futuretalk at Strings At the conference StringMath I am invited to parallel-speak about recent results with D.
Fiorenza, J. Huerta and H. Sati on this question:. In mathematics, homotopy theory is famous for the rich structure that it conjures from minimal input. Generations of mathematicians crank these out with heavy machinery such as the Adams spectral sequence. Each point here has a story associated with it. For instance the three points directly over the numbers 1, 3 and 7, these correspond to the complex, the quaternionic and the octonionic Hopf fibrations.
A filtration know as chromatic homotopy theory proves that the most advanced tools available in homotopy theory, the topological modular forms that host the super-partition function of the heterotic string, are but the second stage of this filtration. Endless wonders lie beyond, waiting to be discovered. The microscope of homotopy theory is the Postnikov-Whitehead tower. Applied to any homotopy type it reveals an infinite Russian doll inner structure. For example the Whitehead tower of the stable orthogonal group reveals an infinite tower of groups whose first stages are named after NS-sector phenomena of string theory: The spin group, the string group, the fivebrane group, the ninebrane group.
Graphics from dcct: horizontally runs the Postnikov tower, vertically the Whitehead tower of the classifying space BO of the stable orthogonal group.
And so forth. In the example of the Whitehead tower of the orthogonal group, the 3-cocycle is the third integral Stiefel-Whitney class, obstructing Spin structure, then next is the 4-cocycle which is the Green-Schwarz anomaly, obstructing String structure, then next is the 8-cocycle which is the Green-Schwarz anomaly of dual heterotic string theory obstructing NS-Fivebrane structure, and so on.
We are talking about pure homotopy theory, and yet stringy terminology keeps appearing. As another example, the string orientation of tmf is an intrinsic homotopy theoretic thing, discovered and studied by mathematicians who know no string theory. And yet it computes the partition function of the heterotic string. What is going on? In order to see more clearly, we should proceed more systematically. We should start at the beginning.