Čech cohomology of infinite projective spaces, flag manifolds, and related spaces

The Witt ring of a complex flag variety describes the interesting – i.e. torsion – part of its topological KO-theory. We show that for a large class of flag varieties, these Witt rings are exterior algebras, and that the degrees of the generators can be determined by Dynkin diagram combinatorics. Besides a few well-studied examples such as full flag varieties and projective spaces, this class includes many flag varieties whose Witt rings were previously unknown, including many flag varieties of exceptional types. In particular, it includes all flag varieties of types G 2 G_2 and F 4 F_4 . The results also extend to flag varieties over other algebraically closed fields.

Let G be a connected, compact Lie group, A (generalized) flag manifold for G is the quotient of G by the centralizer of a torus. Hermitian symmetric spaces (e.g. complex projective spaces and Grassmannians), which are of the form G/C(T) for a circle T ⊂ G, are flag manifolds. The generic examples, though, are flag manifolds of the form G/T for T a maximal torus. The name derives from the manifold U(n)/diagonals of flags C0 = V0 ⊂ V1 ⊂ V2 ⊂ … ⊂ Vn = ℂ n in complex n-space. Flag manifolds enjoy many favorable geometric properties. They can be realized as coadjoint orbits of G, and thus carry an invariant symplectic form. There is also a complex description of flag manifolds as quotients of the complex group Gℂ. The symplectic and complex structures merge nicely: flag manifolds are homogeneous Kahler. Using these two properties—homogeneity and the Kahler condition—we easily compute curvature formulas. For special metrics flag manifolds are Kahler-Einstein. In any metric the Ricci curvature is positive, from which we deduce a vanishing theorem in cohomology. The full flag manifold G/T plays an important role in the representation theory of G a la Borel-Weil-Bott, and in that story the vanishing theorem plays a crucial part.

A geodesic in a Riemannian homogeneous manifold (M = G/K, g) is called a homogeneous geodesic if it is an orbit of a one-parameter subgroup of the Lie group G. We investigate G-invariant metrics with homogeneous geodesics (i.e., such that all geodesics are homogeneous) when M = G/K is a flag manifold, that is, an adjoint orbit of a compact semisimple Lie group G. We use an important invariant of a flag manifold M = G/K, its T-root system, to give a simple necessary condition that M admits a non-standard G-invariant metric with homogeneous geodesics. Hence, the problem reduces substantially to the study of a short list of prospective flag manifolds. A common feature of these spaces is that their isotropy representation has two irreducible components. We prove that among all flag manifolds M = G/K of a simple Lie group G, only the manifold Com(R 2l+2 ) = SO(2l +1)/U(l) of complex structures in R 2l+2 , and the complex projective space CP 2l-1 = Sp(l)/U(1) Sp(l- 1) admit a non-naturally reductive invariant metric with homogeneous geodesics. In all other cases the only G-invariant metric with homogeneous geodesics is the metric which is homothetic to the standard metric (i.e., the metric associated to the negative of the Killing form of the Lie algebra g of G). According to F. Podesta and G.Thorbergsson (2003), these manifolds are the only non-Hermitian symmetric flag manifolds with coisotropic action of the stabilizer.

Consider the iteration of an invertible matrix on the projective space: are the Morse components normally hyperbolic? As far as we know, this was only stablished when the matrix is diagonalizable over the complex numbers. In this article we prove that this is true in the far more general context of an arbitrary element of a semisimple Lie group acting on its generalized flag manifolds: the so called translations on flag manifolds. This context encompasses the iteration of an invertible non-diagonazible matrix on the real or complex projective space, the classical flag manifolds of real or complex nested subspaces and also symplectic grassmanians. Without these tools from Lie theory we do not know how to solve this problem even for the projective space.

We show that there is a ${SL_n}$-stable closed subset of an affine Schubert variety in the infinite dimensional Flag variety (associated to the Kac-Moody group ${\widehat{SL_n}}$) which is a natural compactification of the cotangent bundle to the finite-dimensional Flag variety ${{SL_n/B}}$.

In this paper we study domains in flag manifolds which are bounded in an affine chart and whose projective automorphism group acts co-compactly. In contrast to the many examples in real projective space, we will show that no examples exist in many flag manifolds. Moreover, in the cases where such domains can exist, we show that they satisfy a natural convexity condition and have an invariant metric which generalizes the Hilbert metric. As an application we give some restrictions on the developing map for certain (G,X)-structures.

Numerical matrix eigenvalue methods such as the inverse power iteration or the QR-algorithm can be reformulated as inverse power iterations on homogeneous spaces. In this paper we survey some recent results on controllability properties of the shifted inverse power iteration on flag manifolds. It is shown that the reachable sets are orbits for a semigroup action on the flag manifold. Except for the special case of projective spaces, the algorithm is never controllable. This implies in particular the non-controllability of the shifted QR-algorithm on isospectral matrices. Controllability results for the inverse power iteration on projective space for real or complex shifts are presented, following [20, 22], and a connection with output feedback pole assignability is mentioned. Controllability of the algorithm on Hessenberg flags is shown. This implies controllability of the shifted QR-algorithm on Hessenberg matrices.Key wordsInverse iterationHessenberg matricescontrollabilityreachable setsflag manifoldsQR-algorithm.

The classical Ehresmann-Bruhat order describes the possible degenerations of a pair of flags in a linear space V under linear transformations of V; or equivalently, it describes the closure of an orbit of GL(V acting diagonally on the product of two flag varieties. We consider the degenerations of a triple consisting of two flags and a line, or equivalently the closure of an orbit of GL(V) acting diagonally on the product of two flag varieties and a projective space. We give a simple rank criterion to decide whether one triple can degenerate to another. We also classify the minimal degenerations, which involve not only reflections (i.e., transpositions) in the Weyl group SVSn = dim(V, but also cycles of arbitrary length. Our proofs use only elementary linear algebra and combinatorics.

We study conjugate points along homogeneous geodesics in generalized flag manifolds. This is done by analyzing the second variation of the energy of such geodesics. We also give an example of how the homogeneous Ricci flow can evolve in such way to produce conjugate points in the complex projective space \({\mathbb {C}}P^{2n+1} = \text {Sp}(n+1)/(\text {U}(1)\times \text {Sp}(n))\).

This work is about the algebraic topology of LG/T in particular, the complex cobordism of LG/T where G is a compact semi-simple Lie group. The loop group LG is the group of smooth parametrized loops in G, i.e. the group of smooth maps from the circle S1 into G. Its multiplication is pointwise multiplication of loops. Loop groups turn out to behave like compact Lie groups to a quite remarkable extent. They have Lie algebras which are related to affine Ka?-Moody algebras. The details can be found in [98] and [64]. The class of cohomology theories which we study here are the complex orientable theories. These are theories with a reasonable theory of characteristic classes for complex vector bundles. Complex cobordism is the universal complex orientable theory. This theory has two descriptions. These are homotopy theoretic and geometric. The geometric description only holds for smooth manifolds. Some comments about the structure of this thesis are in order. It is written for a reader with a first course in algebraic topology and some understanding of the structure of compact semi-simple Lie groups and their representations, plus some Hilbert space theory and some mathematical maturity. Some good general references are Kac [60] for Kac-Moody algebra theory, Pressley-Segal [84] for loop groups and their representations, Young [96] for Hilbert space theory, Adams [3] for complex orientable theories, Husemoller [56] and Switzer [93] for fiber bundle theory and topology, Uavenel [87] for Morava K-theories, Lang [74] for the differential topology of infinite dimensional manifolds, Conway [27] for Fredholm operator theory. The organization of this thesis is as follows. Chapter 1 includes all details about Schubert calculus and cohomology of the flag space G/B for Kac-Moody group G. We examine the finite type flag space in section 1. In the section 2, we give some facts and results about Kac-Moody Lie algebras and associated groups and the construction of dual Schubert cocycles on the flag spaces by using the relative Lie algebra cohomology tools. The rest of chapter includes cup product formulas and facts about nil-Hecke rings. Chapter 2 includes the general theory of loop groups. Stratifications and a cell decomposition of Grassmann manifolds and the homogeneous spaces of loop groups are given. In chapter 3, we discuss the calculation of cohomology rings of LG/T. First we describe the root system and Weyl group of LG, then we give some homotopy equivalences between loop grou.ps and homogeneous spaces, and investigate the cohomology ring structures of LSU2/T and ?SU2. Also we prove that BGG-type operators correspond to partial derivation operators on the divided power algebras. In chapter 4, we investigate the topological construction of BGG-type operators, giving details about complex orientable theories, Becker-Gottlieb transfer and a formula of Brumfiel-Madsen. In chapter 5, we develop a version of Quillen's geometric cobordism theory for infinite dimensional separable Hilbert manifolds. For a separable Hilbert manifold X, we prove that this cobordism theory has a graded-group structure under the topological union operation and this theory has push-forward maps. In section 2 of this chapter, we discuss transversal approximations and products, and the contravariant property of this cobordism theory. In section 3, we discuss transversality for finite dimensional fiber bundle. In section 4, we define the Euler class of a finite dimensional complex vector bundle in this cobordism theory and we generalize Bressler-Evens's work on LG/T. In section 6, we prove that strata given in chapter 2 are cobordism classes of infinite dimensional homogeneous spaces. In section 7, we give some examples showing that in certain cases our infinite dimensional theory maps surjectively to complex cobordism.

We extend results of Chang and Ran regarding large dimensional families of immersed curves of positive genus in projective space in two directions. In one direction, we prove a sharp bound for the dimension of a complete family of smooth rational curves immersed into projective space, completing the picture in projective space. In another direction, we isolate the necessary positivity condition on the tangent bundle of projective space used to run the argument, which allows us to rule out large dimensional families of immersed curves of positive genus in generalized flag varieties.

Let (x : y) be homogeneous coordinates on CP . A degree d holomorphic map CP 1 → CP is uniquely determined, up to a constant scalar factor, by N + 1 relatively prime degree d binary forms (f0(x : y) : ... : fN(x : y)). Omitting the condition that the forms are relatively prime we compactify the space of degree d holomorphic maps CP 1 → CP to a complex projective space of dimension (N +1)(d+1)− 1. We denote this compactification of the space of maps by CP d . This construction defines the compactification Πd = Π r i=1CP ni−1 di of the space of degree d = (d1, ..., dr) maps from CP 1 to Π. Composing degree d holomorphic maps from CP 1 to the flag manifold X with the Plucker embedding, we embed the space of such maps into Πd. The closure QMd of this space in Πd is often referred to as the Drinfeld’s compactification of the space of degree d maps from CP 1 to X and will be called the space of quasimaps (following [14]). It is a (generally speaking — singular) irreducible projective variety of complex dimension dimX + 2d1 + ...+ 2dr. The flag manifold is a homogeneous space of the group SLr+1(C) and of its maximal compact subgroup SUr+1. The action of these groups on X lifts naturally to the spaces Π, Πd, and QMd. In addition to this action the spaces Πd and the subspaces QMd carry the circle action induced by the rotation of CP 1 defined by (x : y) 7→ (x : ey). Thus the product group G = S × SUr+1 and its complex version GC = C ∗ × SLr+1(C) act on the quasimap spaces. We will see later that QMd have G-equivariant desingularizations QMd.

Rationally, a map between flag manifolds is seen to be determined up to homotopy by the homomorphism it induces on cohomology. Two algebraic results for cohomology endomorphisms then serve (a) to determine those flag manifolds which have (nontrivial) self-maps that factor through a complex projective space, and (b) for a special class of flag manifolds, to classify the self-maps of their rationalizations up to homotopy.

Rationally, a map between flag manifolds is seen to be determined up to homotopy by the homomorphism it induces on cohomology. Two algebraic results for cohomology endomorphisms then serve (a) to determine those flag manifolds which have (nontrivial) self-maps that factor through a complex projective space, and (b) for a special class of flag manifolds, to classify the self-maps of their rationalizations up to homotopy.

Starting from Borel's description of the mod-2 cohomology of real flag manifolds, we give a minimal presentation of the cohomology ring for semi complete flag manifolds $F_{k,m}:=F(1,\ldots,1,m)$ where $1$ is repeated $k$ times. The information is used in order to estimate Farber's topological complexity of these spaces when $m$ approaches (from below) a 2-power. In particular, we get almost sharp estimates for $F_{2,2^e-1}$ which resemble the known situation for the real projective spaces $F_{1,2^e}$. Our results indicate that the agreement between the topological complexity and the immersion dimension of real projective spaces no longer holds for other flag manifolds. More interestingly, we also get corresponding results for the $s$-th (higher) topological complexity of these spaces. Actually, we prove the surprising fact that, as $s$ increases, the estimates become stronger. Indeed, we get several full computations of the higher motion planning problem of these manifolds. This property is also shown to hold for surfaces: we get a complete computation of the higher topological complexity of all closed surfaces (orientable or not). A homotopy-obstruction explanation is included for the phenomenon of having a cohomologically accessible higher topological complexity even when the regular topological complexity is not so accessible.