Phillip E. Parker

A User’s Guide to Algebraic Topology

Preface. Introduction and Overview. 1. Basics of Extension and Lifting Problems. 2. Up to Homotopy is Good Enough. 3. Homotopy Group Theory. 4. Homology and Cohomology Theories. 5. Examples in Homology and Cohomology. 6. Sheaf and Spectral Theories. 7. Bundle Theory. 8. Obstruction Theory. 9. Applications. A: Algebra. B: Topology. C: Manifolds and Bundles. D: Tables of Homotopy Groups. E: Computational Algebraic Topology. Bibliography. Index.

Pseudoconvexity and geodesic connectedness

SummaryLet M be a smooth manifold with a linear connection. If M is pseudoconvex, disprisoning, and has no conjugate points, then each pair of points of M is joined by at least one geodesic.

On some theorems of Geroch and Stiefel

Modern proofs of two theorems of Geroch (spinor spacetimes and globally hyperbolic spacetimes are parallelizable) and a theorem of Stiefel (orientable 3‐manifolds are parallelizable) are given, using the computationally efficient obstruction theory of algebraic topology. These techniques also easily show that, in fact, Geroch’s second theorem is a corollary of Stiefel’s theorem.

The space of geodesics

Let M be a manifold with linear connection ▽. The space G(M) of all geodesics of M may be given a topological structure and may be realized as a quotient space of the reduced tangent bundle of M. The space G(M) is a T1 space iff the image of each geodesic is a closed subset of M. It is Hausdorff iff each tangentially convergent sequence of geodesics converges in the Hausdorff limit sense to the limit geodesic. If M has no conjugate points and G(M) is Hausdorff, then M is geodesically connected.

Sectional curvature and tidal accelerations

Sectional curvature is related to tidal accelerations for small objects of nonzero rest mass. Generically, the magnification of tidal accelerations due to high speed goes as the square of the magnification of energy. However, some space‐times have directions with bounded increases in tidal accelerations for relativistic speeds. These investigations also yield a characterization of null directions that fail to satisfy the generic condition used in singularity theorems. For Ricci flat four‐dimensional space‐times, tidally nondestructive directions are characterized as repeated principal null directions.

Lattices and Periodic Geodesics in Pseudoriemannian 2-step Nilpotent Lie Groups

We give a basic treatment of lattices Γ in these groups. Certain tori TF and TB provide the model fiber and the base for a submersion of Γ\N. This submersion may not be pseudoriemannian in the usual sense, because the tori may be degenerate. We then begin the study of periodic geodesics in these compact nilmanifolds, obtaining a complete calculation of the period spectrum of certain flat spaces.

Spaces of geodesics: products, coverings, connectedness

We continue our study of the space of geodesics of a manifold with linear connection. We obtain sufficient conditions for a product to have a space of geodesics which is a manifold. We investigate the relationship of the space of geodesics of a covering manifold to that of the base space. We obtain sufficient conditions for a space to be geodesically connected in terms of the topology of its space of geodesics.

Pseudoconvex and disprisoning homogeneous sprays

The pseudoconvex and disprisoning conditions for geodesics of linear connections are extended to the solution curves of general homogeneous sprays. The main result is that pseudoconvexity and disprisonment are jointly stable in the fine topology on the space of all homogeneous sprays of any degree of homogeneity.

SYMMETRIES OF SECTIONAL CURVATURE ON 3-MANIFOLDS

In dimension three, there are only two signatures of metric tensors: Lorentzian and Riemannian. We find the possible pointwise symmetry groups of Lorentzian sectional curvatures considered as rational functions, and determine which can be realized on naturally reductive homogeneous spaces. We also give some examples. MSC (1991): Primary 53C50; Secondary 53B30, 53C30. 1Supported by Project XUGA8050189, Xunta de Galicia, Spain. 2On leave from Math. Dept., Wichita State Univ., Wichita KS 67260, U.S.A., pparker@twsuvm.uc.twsu.edu 3Partially supported by DGICYT-Spain.