Homology and cohomology theories on manifolds

We study generalized homology and cohomology theories on the category of manifolds. Ordinary (co-)homology including the cup product is characterized axiomatically

Exotic Spaces in Quantum Gravity I: Euclidean Quantum Gravity in Seven Dimensions

It is well known that in four or more dimensions, there exist exotic manifolds; manifolds that are homeomorphic but not diffeomorphic to each other. More precisely, exotic manifolds are the same topological manifold but have inequivalent differentiable structures. This situation is in contrast to the uniqueness of the differentiable structure on topological manifolds in one, two and three dimensions. As exotic manifolds are not diffeomorphic, one can argue that quantum amplitudes for gravity formulated as functional integrals should include a sum over not only physically distinct geometries and topologies but also inequivalent differentiable structures. But can the inclusion of exotic manifolds in such sums make a significant contribution to these quantum amplitudes? This paper will demonstrate that it will. Simply connected exotic Einstein manifolds with positive curvature exist in seven dimensions. Their metrics are found numerically; they are shown to have volumes of the same order of magnitude. Their contribution to the semiclassical evaluation of the partition function for Euclidean quantum gravity in seven dimensions is evaluated and found to be nontrivial. Consequently, inequivalent differentiable structures should be included in the formulation of sums over histories for quantum gravity.Comment: AmsTex, 23 pages 5 eps figures; replaced figures with ones which are hopefully viewable in pdf forma

A classification of smooth embeddings of 3-manifolds in 6-space

We work in the smooth category. If there are knotted embeddings S^n\to R^m, which often happens for 2m<3n+4, then no concrete complete description of embeddings of n-manifolds into R^m up to isotopy was known, except for disjoint unions of spheres. Let N be a closed connected orientable 3-manifold. Our main result is the following description of the set Emb^6(N) of embeddings N\to R^6 up to isotopy. The Whitney invariant W : Emb^6(N) \to H_1(N;Z) is surjective. For each u \in H_1(N;Z) the Kreck invariant \eta_u : W^{-1}u \to Z_{d(u)} is bijective, where d(u) is the divisibility of the projection of u to the free part of H_1(N;Z). The group Emb^6(S^3) is isomorphic to Z (Haefliger). This group acts on Emb^6(N) by embedded connected sum. It was proved that the orbit space of this action maps under W bijectively to H_1(N;Z) (by Vrabec and Haefliger's smoothing theory). The new part of our classification result is determination of the orbits of the action. E. g. for N=RP^3 the action is free, while for N=S^1\times S^2 we construct explicitly an embedding f : N \to R^6 such that for each knot l:S^3\to R^6 the embedding f#l is isotopic to f. Our proof uses new approaches involving the Kreck modified surgery theory or the Boechat-Haefliger formula for smoothing obstruction.Comment: 32 pages, a link to http://www.springerlink.com added, to appear in Math. Zei

Synthetic Aperture Radar Processing in 2D and 3D Space Using Raised Cosine Transmitted Pulses

The objective of this paper is to develop two and three dimensional simulated radar images using transmitted raised cosine pulse signals. Input data consists of elevation database information and an array of radar source locations. The transmitted signal is assumed to be an isotropic signal and interrogates each elevation point or scattering center in the database. Range time intensity methodologies are developed and simulated. In these simulations the raised cosine transmitted signal is transmitted and received at multiple locations on a circular path around a target. For each radar location the return data is processed to obtain range resolved signals. This data is then stacked and integrated to generate a two dimensional image. Three dimensional images are developed by experimenting with a variety of radar source patterns around a target center. In these experiments the radar source patterns tested will be concentric circular paths, single circular paths, random source locations, two parallel lines, and a grid. In these simulations, range return data will be stacked and slices of two dimensional returns will be processed at different heights to develop a stacked three dimensional image. These simulations are executed with the assumption that the radar source is moving as in traditional synthetic aperture radar applications. However, the methodology employed can also be utilized in inverse synthetic aperture radar applications in which the radar is stationary and the targets are moved

Immersions of punctured 4-manifolds

Motivated by applications to pulling back quantum cellular automata from one manifold to another, we study the existence of immersions between certain smooth 4-manifolds. We show that they lead to a very interesting partial order on closed 4-manifolds