Cohomology of quaternionic projective space pdf

Yumi boote university of manchester the integral cohomology of the symmetric square of quaternionic projective space2014 6. Z are given by jesus gonz alez and peter landweber in 2012. All these definitions extend naturally to the case where k is a division ring. Denote by k a maximal com pact subgroup of the real lie group gr. That is just enough associativity to construct the projective plane, but not enough to construct projective 3space. The quaternionic flag manifold the goal of this section is to give a few alternative presentations of the manifold fl n h, which will be used later. This work was supported by the grant msm0 021622409 of the.

Submanifolds of dimension in a quaternionic projective. Lx hs1with the negative cyclic homology of the singular cochain algebra c. The existence and uniqueness theorems are postponed to the final chapters. We discuss how complex projective space for k k the real numbers or the complex numbers equipped with their euclidean metric topology is a topological manifold and naturally carries the structure of a smooth manifold prop. Vanishing theorem for cohomology groups of c2selfdual bundles on quaternionic kahler manifolds yasuyuki nagatomo department of mathematics, tokyo metropolitan university, minamiohsawa ll, hachiojishi, tokyo 19203, japan. Almost quaternionic structures on eightmanifolds martin cadek and jiri vanzura received march 1, 1996 1.

Pdf combinatorial 8manifolds having cohomology of the. On the geometric weight system of topological actions on. Ishihara, notes on quaternion kaehler manifolds, to appear 3 s. Applications of characteristic classes and milnors exotic. In this paper the word manifold will always mean oriented compact c manifold. We offer a solution for the complex and quaternionic projective. Since we are going to use both the normal jacobi operator and the usual jacobi operator, we will introduce the following notation. Maps to spaces in the genus of infinite quaternionic projective space. String homology of spheres and projective spaces 311 insection 2we introduce a cohomology theory for frobenius algebras that is dual to negative cyclic homology. Equivariant cohomology of quaternionic flag manifolds.

Schwarzenberger sc, hi has shown the fact that a kdimensional fvector bundle v over fpn for f r or c is stably equivalent to a whitney sum of k fline bundles if v is extendible, that is, if v is the restriction of a fvector bundle over fpm for any 111 n. But, for a higherdimensional case, further information about cohomology groups is needed for an analogue of a. Homology of quaternionic projective space topospaces. In this lecture, we will address the question of how canonical this structure is. Let pm denot thee orbit space whic ish obtained by identifying point osf sm with their transform ths undee actionr o f s. Configuration spaces, the octonionic projective plane, and.

This extends a similar result, including the method of proof, already known for the nonembeddings of real and complex projective planes. In their paper 2, 3, kwon and pak had studied submanifolds of dimension isometrically immersed in a quaternionic projective space and proved the following theorem as a quaternionic analogy to theorems given in 14, 15, which are natural extensions of theorems proved in to the case of submanifolds with dimension and also extensions of. If m0 m1 is a sequence of omodules, then limh qx,m k. In this paper we completely classify the homogeneous twospheres, especially, the minimal homogeneous ones in the quaternionic projective space hpn. Quaternionic projective space and homotopy groups of spheres.

An explicit growth condition for middledegree cuspidal cohomology of arithmetically defined quaternionic hyperbolic nmanifolds harald grobner abstract. The integral cohomology of the symmetric square of. I think the real reason that the cayley projective plane exists is because any subalgebra of the octonions that is generated by 2 elements is associative. Self maps of projective spaces 327 both of these theorems can be proved using first principles in algebraic topology and so their proofs are omitted. Applications of characteristic classes and milnors exotic spheres roisin dempsey braddell. Cohomology of groups and mod ules, cambridge studies in advanced mathematics, cambridge university press 1991. It presents recent results by top notch experts, and is intended primarily for researchers and graduate students working. The problem of computing the integral cohomology ring of the symmetric square of a topological space has long been of interest, but limited progress has been made on the general case until recently.

This allows us to dualize jones theorem10identifying h. X as a representation of the modp steenrod algebra a p. Kuhnel, each of which is a cohomology quaternionic projective plane, can be combinatorially embedded in the euclidean space e12, though they have tight polyhedral embeddings in e14. Introduction spn is the group of the quaternionic linear automorphisms acting from the left on a right quaternionic ndimensional vector space preserving a positive definite hermitian form on it. Real hypersurfaces in quaternionic projective space. The homology of quaternionic projective space is given as follows. Quaternionic projective space, o ctonionic pro jective plane, free loop spac e, integral loop homo logy, batalinvilkovisky algebra. Categorical decomposition techniques in algebraic topology. This topological space is the main object of study in this paper. In this paper, we shall classify conformal nonsuperminimal harmonic twotori in a 2 or 3dimensional quaternionic projective space, which are not always covered by primitive harmonic twotori of nite type. Ishihara, integral formulas and their applications in quaternion kaehle manifolds, to appear. We offer a solution for the complex and quaternionic projective spaces pn, by utilising their rich geometrical structure. Vanishing theorem for cohomology groups of c2selfdual.

Quaternionic projective space lecture 34 july 11, 2008 the threesphere s3 can be identi. The complex projective line is also called the riemann sphere. The book consists of articles at the frontier of current research in algebraic topology. Unless otherwise specified, all homology and cohomology is taken with integral coefficients, and for m an manifold, me hnm, dm will denote the orientation class of m. This paper is devoted to make a systematic study of real hypersurfaces of quaternionic projective space using focal set theory. On the geometric weight system of topological actions on cohomology quaternionic projective spaces. Stably extendible vector bundles over the quaternionic. Show that the cw structure of hpn consists of only one. Introduction the harmonic twospheres in ndimensional quaternionic projective space hpn. The space h n is a h module with respect to scalar multiplication from the left. The betti numbers of quaternionic projective space are thus for with and elsewhere.

Nevertheless, the cohomology of a space, which is obtained by dualizing its simplicail chain complex, carries important additional structure. In all three cases real, complex, and quaternionic, the collapsing of the space skd k 0, 1,3 only occurs over the boundary of polytope, hence the corresponding space has a natural smooth structure over the interior of p. X the graded abelian group of singular chains on x, so snx is the free abelian group generated by the singular nsimplices. The cohomology of quaternionic hyperplane complements. If n is one or two, a projective space of dimension n is called a projective line or a projective plane, respectively. The cohomology of quaternionic hyperplane complements william schlieper. Third type is new and in its study the first example of proper quaternion crsubmanifold appears.