Odd Spheres Research Articles

Elliptic aspects of statistical mechanics on spheres

Our earlier results on the temperature inversion properties and the ellipticization of the finite temperature internal energy on odd spheres are extended to orbifold factors of odd spheres and then to other thermodynamic quantities, in particular, to the specific heat. The behavior under modular transformations is facilitated by the introduction of a modular covariant derivative and it is shown that the specific heat on any odd sphere can be expressed in terms of just three functions. It is also shown that the free energy on the circle can be written elliptically.

THE BOUSFIELD–KAN SPECTRAL SEQUENCE FOR MORAVA $K$-THEORY

AbstractWe construct a spectral sequence converging to the $E_{2}$-term of the Bousfield–Kan spectral sequence (BKSS) for a wide variety of homology theories. Using this, the $E_{2}$-term of the BKSS based on $K(1)$-theory for the odd spheres is computed and the unstable $K(1)$-completion is computed.

Fredholm modules for quantum Euclidean spheres

The quantum Euclidean spheres, S q N−1 , are (noncommutative) homogeneous spaces of quantum orthogonal groups, SO q ( N). The ∗-algebra A( S N−1 q ) of polynomial functions on each of these is given by generators and relations which can be expressed in terms of a self-adjoint, unipotent matrix. We explicitly construct complete sets of generators for the K-theory (by nontrivial self-adjoint idempotents and unitaries) and the K-homology (by nontrivial Fredholm modules) of the spheres S q N−1 . We also construct the corresponding Chern characters in cyclic homology and cohomology and compute the pairing of K-theory with K-homology. On odd spheres (i.e., for N even) we exhibit unbounded Fredholm modules by means of a natural unbounded operator D which, while failing to have compact resolvent, has bounded commutators with all elements in the algebra A( S N−1 q ).

Higher dimensional geometries related to fuzzy odd-dimensional spheres

We study $SO(m)$ covariant Matrix realizations of $ \sum_{i=1}^{m} X_i^2 = 1 $ for even $m$ as candidate fuzzy odd spheres following hep-th/0101001. As for the fuzzy four sphere, these Matrix algebras contain more degrees of freedom than the sphere itself and the full set of variables has a geometrical description in terms of a higher dimensional coset. The fuzzy $S^{2k-1} $ is related to a higher dimensional coset $ {SO(2k) \over U(1) \times U(k-1)}$. These cosets are bundles where base and fibre are hermitian symmetric spaces. The detailed form of the generators and relations for the Matrix algebras related to the fuzzy three-spheres suggests Matrix actions which admit the fuzzy spheres as solutions. These Matrix actions are compared with the BFSS, IKKT and BMN Matrix models as well as some others. The geometry and combinatorics of fuzzy odd spheres lead to some remarks on the transverse five-brane problem of Matrix theories and the exotic scaling of the entropy of 5-branes with the brane number.

On the Polarization of Unstable D0-Branes into Noncommutative Odd Spheres

We consider the polarization of unstable type IIB D0-branes in the presence of a background five-form field strength. This phenomenon is studied from the point of view of the leading terms in the non-abelian Born Infeld action of the unstable D0-branes. The equations have SO(4) invariant solutions describing a non-commutative 3-sphere, which becomes a classical 3-sphere in the large N limit. We discuss the interpretation of these solutions as spherical D3-branes. The tachyon plays a tantalizingly geometrical role in relating the fuzzy S^3 geometry to that of a fuzzy S^4.

Approximating rational spaces with elliptic complexes and a conjecture of Anick

An elliptic space is one whose rational homotopy and rational cohomology are both nite dimensional. David Anick conjectured that any simply connected nite CW-complex S can be realized as the k-skeleton of some elliptic complex as long as k> dimS. A functorial version of this conjecture due to McGibbon is that for any n there exists an elliptic complex En and an n-equivalence S ! En. In fact, this is equivalent to its Eckmann-Hilton dual, which we prove in the rational category for a small class of simply connected spaces. Moreover, we construct the n-equivalence in such a way that the homotopy bre is, rationally, a product of a nite number of odd spheres.

Rational Classification of Simple Function Space Components for Flag Manifolds

AbstractLetM(X, Y) denote the space of all continuous functions between X and Y and Mƒ(X, Y) the path component corresponding to a given map ƒ : X → Y. When X and Y are classical flag manifolds, we prove the components of M(X, Y) corresponding to “simple” maps ƒ are classified up to rational homotopy type by the dimension of the kernel of ƒ in degree two cohomology. In fact, these components are themselves all products of flag manifolds and odd spheres.

On the cohomology of freep-torus actions

1. Preliminaries. Let p be a prime, ;gp:= 7Z/(p). We shall denote 2gp by G, if we mean the cyclic group, and by k, if we mean the coefficient field. For p = 2 the group ring k [G] of G is just an exterior algebra. In general it is a quotient of a polynomial ring of: k[x]/(xP)~ k[G] "2 ~---~1--g, where g denotes a generator of G. The representation of the odd spheres S 2"+~ with canonical free G-action and the universal G-space E o, their projective limit (the "infinite sphere"), as G-C W complexes (cf. [16, p. 100]), leads to the following cellular G-cochain complexes of Y: = S am+ 1, Ea and B~, the classifying space of G, which in fact is an "infinite lens space"

The K-theory localization of loops on an odd sphere and applications

K-theory localization of an unstable odd sphere is constructed by Mahowald and Thompson in [8]. Earlier work on unstable K-theory localization includes Bousfield’s construction of the K-theory localization of infinite loop spaces in [4] and Mislin’s construction of the K-theory localization of Eilenberg-MacLane spaces in [13]. The con- struction of Mislin demonstrates that the unstable K-theory localization functor does not commute with taking loops on a space, nor does it preserve the connectivity of a space. This being the case, we note that the K-theory localization of loops on an odd sphere cannot be directly given by knowing the K-theory localization of an odd sphere. In recent work Bousfield avoids this difficulty by defining a new vi localization functor which behaves well with respect to fibrations and then shows that it is closely related to the unstable K-theory localization functor. In this paper a more direct approach will be taken to construct the K-theory localiza- tion of loops on an odd sphere and double loops on an odd sphere. The construction and proof is a generalization of the results of Mahowald and Thompson in [IS]. As an application, these results will be used to construct the K-theory localization of the fiber of the Hopf map. Construction and Results: We will begin by summarizing the construction of the K-theory localization of loops on an odd sphere done in [S]. By [14] the localization with respect to real and complex K-theory give the same result. Throughout this paper K-theory will refer to complex K-theory. For simplicity, we will take the localization with respect to KCpJ at each prime separately, where K,,, denotes p local complex K-theory. These localiza- tions may then be pieced together to give the K-theory localization. Let F, denote the homotopy fiber of the Snaith map where QX = C2= C”X, q = 2p - 2, and B, denotes a stunted BXP, localized at p with bottom cell in dimension m. The inclusion map ii :S2”+’ -+ QS2”+’ lifts to a map ji:S”‘+i -+ F,. Mahowald and Thompson show in [S] that the map jr induces an isomor- phism in p local K-theory. This is the first step in showing that their construction is the K-theory localization of S2”+ ‘. Now let G, denote the homotopy fiber of the localized Snaith map

The spinor heat kernel in maximally symmetric spaces

The heat kernelK(x, x′, t) of the iterated Dirac operator on anN-dimensional simply connected maximally symmetric Riemannian manifold is calculated. On the odd-dimesional hyperbolic spacesK is a Minakshisundaram-DeWitt expansion which terminates to the coefficienta N−1)/2 and is exact. On the odd spheres the heat kernel may be written as an image sum of WKB kernels, each term corresponding to a classical path (geodesic). In the even dimensional case the WKB approximation is not exact, but a closed form ofK is derived both in terms of (spherical) eigenfunctions and of a “sum over classical paths.” The spinor Plancherel measure μ(λ) and ζ function in the hyperbolic case are also calculated. A simple relation between the analytic structure of μ onH N and the degeneracies of the Dirac operator onS N is found.

Some relations between Toeplitz and singular integral operators on odd spheres

Some relations between Toeplitz and singular integral operators on odd spheres

On the space of free loops of an odd sphere

On the space of free loops of an odd sphere

Some Calculations In the Unstable Adams-Novikov Spectral Sequence

The unstable Adams-Novikov spectral sequence for a space X is a sequence of groups {Er(X)}9 r = 2, 3,..., which converge to the homotopy groups of X, and whose £2term depends on the complex cobordism groups of X. We investigate this spectral sequence when X is the infinite special unitary group SU, or one of the finite groups SU(n), or when X is an odd sphere S2+1. The reader is referred to [2] for the construction and properties of the unstable Adams-Novikov spectral sequence. For some purposes, it is convenient to localize at a prime p, in which case the complex cobordism homology theory, based on the spectrum MU, is replaced by Brown-Peterson homology, based on the spectrum BP. We then have a useful spectral sequence with many of the properties of the stable Adams-Novikov spectral sequence. Namely, the nitrations are less than or equal to the filtrations in the unstable Adams spectral sequence based on mod-p homology. When X is a space for which H*(X; BP) is free over the coefficient ring n*(BP) and cofree as a coalgebra, then the E2-term is isomorphic to an Ext group in an abelian category (see §2; also [2, § 7]). Furthermore, this Ext group may be computed as the homology of an unstable cobar complex which we describe explicitly in Section 2. In particular, these considerations apply to the cases X = SU, X = SU(ri), or X = S2n+1. We first consider the situation where X is a p-local H-space with torsion-free homotopy and torsion-free homology. The results of Wilson [10] and the

Torus Actions on a Cohomology Product of Three Odd Spheres

Torus Actions on a Cohomology Product of Three Odd Spheres

Torus actions on a cohomology product of three odd spheres

The main purpose of this paper is to describe how a torus group may act on a space, X X , whose rational cohomology ring is isomorphic to that of a product of three odd-dimensional spheres, in such a way that the fixed point set is nonempty, and X X is not totally nonhomologous to zero in the associated X X -bundle, X T → B T {X_T} \to {B_T} . In the first section of the paper some general results on the cohomology theory of torus actions are established. In the second section the cohomology theory of the above type of action is described; and in the third section the results of the first two sections are used to prove a Golber formula for such actions, which, under certain conditions, bears an interesting interpretation in terms of rational homotopy.

Torus actions on a product of two odd spheres

THE OBJECT of this paper is to describe certain methods which may be applied to the study of arbitrary topological torus actions, and to apply these methods to actions on a product of two odd spheres. We are motivated by certain recent papers of Wu-Yi Hsiang ([4], [5] and others), which study actions of a compact connected Lie group on Euclidean spaces, spheres, and complex projective spaces by restrictin, 0 an action to a maximal torus of the group and applying certain known results concernin g torus actions on these spaces. It is hoped that the methods and results of this paper may be useful in obtaining results about actions of compact connected Lie groups on other spaces.

Some Whitehead Products on Odd Spheres

Some Whitehead Products on Odd Spheres

Some Whitehead products on odd spheres

Some Whitehead products on odd spheres