Peter Zvengrowski

Kinks in general relativity

The problem of classifying topologically distinct general relativistic metrics is discussed. For a wide class of parallelizable space-time manifolds it is shown that a certain integer-valued topological invariant n always exists, and that quantization when n is odd will lead to spinor wave functionals.

The cohomology ring of a class of Seifert manifolds

Abstract The cohomology groups of the Seifert manifolds are well known. In this article a method is given to compute the cup products in the cohomology ring of any orientable Seifert manifold whose associated orbit surface is S 2 , and for any coefficients. In particular the Z /2 cohomology ring is completely determined. This is applied to determine the existence of degree 1 maps from the Seifert manifold to R P 3 , and to the Lusternik–Schnirelmann category.

Spin in kink-type field theories

This paper studies some classical three-dimensional field theories for which the ranges of the field variables are a 3-sphere, a 2-sphere, the symplectic group,Sp(n), the special orthogonal group,SO(3), and theS4,1 space of general relativistic metrics. The main result is the proof that these theories admit half-odd-integer spin, so that the 1-kink states are classical analogs of fermion states.

The cohomology ring of the orientable Seifert manifolds, II

Abstract The cohomology ring of an arbitrary orientable Seifert manifold is computed with Z /p coefficients for any prime p . In some cases the cohomology rings are given with Z /p s coefficients. These results will be used to compute the abelian Witten–Reshetikhin–Turaev type invariants and Dijkgraaf–Witten invariants for some classes of Seifert manifolds in a later paper. Finally, necessary and sufficient conditions for the existence of a degree one map from an orientable Seifert manifold into a lens space are given.

TYPE OF 3-MANIFOLDS AND ADDITION OF RELATIVISTIC KINKS

The type of a closed, connected, orientable 3-manifold M was first considered in the classification of relativistic kinks over the space-time 4-manifold M × R. In this paper theorems are developed relating the type of M to H1 (M; Z), which lead to the determination of the type of large families of 3-manifolds. The relation of type to connected sum is established, and the connected sum is also used to define addition of kinks. The kink addition is related to the sum of the kink numbers, as well as addition in the bordism group Ω3(P3).

Stable parallelizability of partially oriented flag manifolds II

In the first paper with the same title the authors were able to determine all partially oriented flag manifolds that are stably parall elizable or parallelizable, apart from four infinite families that were undecided. Here, using more delicate techniques (mainly K-theory), we settle these previously undecided families and show that none of the manifolds in them is stably parallelizable, apart from o ne 30-dimensional manifold which still remains undecided.

Kink metrics in (2+1)‐dimensional space‐time

A homotopy classification scheme is developed for Lorentz metrics that are defined on (2+1)‐dimensional space‐times of various topologies. The ‘‘kink number,’’ in the sense of the Finkelstein–Misner kink number in higher dimensions, is discussed, and examples are presented of kink metrics that satisfy the Einstein field equations in 2+1 dimensions.

Skewness of r-fields on spheres

INTRODUCTION LET S’ = {x : x E R’, [[xl] = 1) and x = ( xl, x2. x3, x4) E 9. Setting u,(x) = (-x2, xl, -x4, x3 defines a unit vector which depends continuously on x and is always orthogonal to x, i.e., a vector field on S3 consisting of unit vectors (a “l-field”). Another such l-field would clearly be uz(x) = (-xz, XI, x4. -x3, and it can be shown that u, and uz are not homotopic (through l-fields). For details of this proof, as well as the other assertions in this paragraph, we refer the reader to [12]. Both u, and uz are linear in the sense that the co-ordinate functions are linear, and of the countably many homotopy classes of l-fields on S’, any linear l-field turns out to be homotopic to u, or oz. An example of a l-field on S3 homotopic to neither u, nor uz and hence to no linear field is given by

The cohomology algebras of orientable Seifert manifolds and applications to Lusternik-Schnirelmann category

This note gives a complete description of the cohomology algebra of any orientable Seifert manifold with Z/p coefficients, for an arbitrary prime p. As an application, the existence of a degree one map from an orientable Seifert manifold onto a lens space is completely determined. A second application shows that the Lusternik–Schnirelmann category for a large class of Seifert manifolds is equal to 3, which in turn is used to verify the Ganea conjecture for these Seifert manifolds.

Vertex embeddings of regular polytopes

Abstract The question of when one regular polytope (finite, convex) embedds in the vertices of another, of the same dimension, leads to a fascinating interplay of geometry, combinatorics, and matrix theory, with further relations to number theory and algebraic topology. This mainly expository paper is an account of this subject, its history, and the principal results together with an outline of their proofs. The relationships with other branches of mathematics are also explained.