Nitu Kitchloo

Compatible complex structures on symplectic rational ruled surfaces

In this article, we study the topology of the spaceIω of complex structures compatible with a fixed symplectic form ω, using the framework of Donaldson. By comparing our analysis of the spaceIω with results of McDuff on the spaceJω of compatible almost complex structures on rational ruled surfaces, we find that Iω is contractible in this case. We then apply this result to study the topology of the symplectomorphism group of a rational ruled surface, extending results of Abreu and McDuff.

Finite loop spaces are manifolds

One of the motivating questions for surgery theory was whether every finite H:space is homotopy equivalent to a Lie group. This question was answered in the negative by Hilton and Roitberg s discovery of some counterexamples [18]. However, the problem remained whether every finite H-space is homotopy equivalent to a closed, smooth manifold. This question is still open, but in case the H-space admits a classifying space we have the following theorem.

On complexes equivalent to S3-bundles over S4

S3-bundles over S4 have played an important role in topology and geometry since Milnor showed that the total spaces of such bundles with Euler class ±1 are manifolds homeomorphic to S7 but not always diffeomorphic to it. In 1974, Gromoll and Meyer exhibited one of these spheres (a generator in the group of homotopy 7-spheres) as a double coset manifold i.e. a quotient of Sp(2) hence showing that it admits a metric of nonnegative curvature (cf. [6]). Until recently, this was the only exotic sphere known to admit a metric of nonnegative sectional curvature. Then in [7], K. Grove and W. Ziller constructed metrics of nonnegative curvature on the total space of S3-bundles over S4. They also asked for a classification of these bundles up to homotopy equivalence, homeomorphism and diffeomorphism. These questions have been addressed in many papers such as [12], [11], [15] and more recently in [3]. In this paper we attempt to fill the gap in the previous papers; we consider the problem of determining when a given CW complex is homotopy equivalent to such a bundle. The problem was motivated by [7]: the Berger space, Sp(2)/Sp(1), is a 7-manifold that has the cohomology ring of an S3-bundle over S4, but does it admit the structure of such a bundle? The fact that it cannot be a principal S3-bundle over S4 is straightforward and is proved in [7].

The ER(n)-cohomology of BO(q) and real Johnson–Wilson orientations for vector bundles

Using the Bockstein spectral sequence developed previously by the authors, we compute the ring ER(n) ∗ (BO(q)) explicitly. We then use this calculation to show that the ring spectrum MO[2 n+1 ]i sER(n)-orientable (but not ER(n +1 )-orientable), whereMO[2 n+1 ]i s def ined as the Thom spectrum for the self-map of BO given by multiplication by 2 n+1 .

On fibrations related to real spectra

We consider real spectra, collections of Z=(2)–spaces indexed over Z Z withcompatibility conditions. We produce fibrations connecting the homotopy fixedpointsandthespacesinthesespectra. Wealsoevaluatethemapwhichistheanalogueof the forgetful functor from complex to reals composed with complexification.Our first fibration is used to connect the real 2

Elliptic Cohomology: Thom prospectra for loopgroup representations

This is very much an account of work in progress. We sketch the construction of an Atiyah dual (in the category of T-spaces) for the free loopspace of a manifold; the main technical tool is a kind of Tits building for loop groups, discussed in detail in an appendix. Together with a new local- ization theorem for T-equivariant K-theory, this yields a construction of the elliptic genus in the string topology framework of Chas-Sullivan, Cohen-Jones, Dwyer, Klein, and others. We also show how the Tits building can be used to construct the dualizing spectrum of the loop group. Using a tentative notion of equivariant K-theory for loop groups, we relate the equivariant K-theory of the dualizing spectrum to recent work of Freed, Hopkins and Teleman.

A homotopy construction of the adjoint representation for Lie groups

Let G be a compact, simply-connected, simple Lie group and T G a maximal torus. The purpose of this paper is to study the connection between various brations over BG (where G is a compact, simply-connected, simple Lie group) associated to the adjoint representation and homotopy colimits over poset categories C, hocolimCBGI where GI are certain connected maximal rank subgroups of G.

Real Structures and Morava K-Theories

We show that real k-structures coincide for k = 1, 2 on all formal groups for which multiplication by 2 is an epimorphism. This enables us to give explicit polynomial generators for the Morava K(n)-homology of BSpin and BSO for n = 1, 2.

COHOMOLOGY SPLITTINGS OF STIEFEL MANIFOLDS

The complex Stiefel manifolds admit a stable decomposition as Thom spaces of certain bundles over Grassmannians. The purpose of the paper is to identify the splitting in any complex oriented cohomology theory.

Landweber flat real pairs and ER(n)-cohomology

Abstract We take advantage of the internal algebraic structure of the Bockstein spectral sequence converging to E R ( n ) ⁎ ( p t ) to prove that for spaces Z that are part of Landweber flat real pairs with respect to E ( n ) (see Definition 2.9), the cohomology ring E R ( n ) ⁎ ( Z ) can be obtained from E ( n ) ⁎ ( Z ) by base change. In particular, our results allow us to compute the Real Johnson–Wilson cohomology of the Eilenberg–MacLane spaces Z = K ( Z , 2 m + 1 ) , K ( Z / 2 q , 2 m ) , K ( Z / 2 , m ) for any natural numbers m and q, as well as connective covers of BO: BO , BSO , BSpin and BO 〈 8 〉 .