Unoriented geometric functors

Farrell and Hsiang noticed that the geometric surgery groups defined By Wall, Chapter 9, do not have the naturality Wall claims for them. They were able to fix the problem by augmenting Wall's definitions to keep track of a line bundle. The definition of geometric Wall groups involves homology with local coefficients and these also lack Wall's claimed naturality. One would hope that a geometric bordism theory involving non-orientable manifolds would enjoy the same naturality as that enjoyed by homology with local coefficients. A setting for this naturality entirely in terms of local coefficients is presented in this paper. Applying this theory to the example of non-orientable Wall groups restores much of the elegance of Wall's original approach. Furthermore, a geometric determination of the map induced by conjugation by a group element is given.Comment: 12 pages, LaTe

Codimension one spheres which are null homotopic

This paper classifies embedded, codimension-one spheres which are null homotopic. This information is used to show that all null homotopic, immersed codimension-one spheres which are taut in the sense of Terng and Thorbergsson are actually distance spheres.Comment: 6 page

Discussion of a physical optics method and its application to absorbing smooth and slightly rough hexagonal prisms

Three different mathematical solutions of a physical optics model for far field diffraction by an aperture due to Karczewski and Wolf are discussed. Only one of them properly describes diffraction by an aperture and can, by applying Babinet's principle, be used to model diffraction by the corresponding plane obstacle, and by further approximation, diffraction by a particle. Studying absorbing scatterers allows a closer investigation of the external diffraction component because transmission is negligible. The physical optics model has been improved on two aspects: (i) To apply the diffraction model based on two-dimensional apertures more accurately to three-dimensional objects, a size parameter dependent volume obliquity factor is introduced, thus reducing the slightly overestimated side scattering computed for three-dimensional objects. (ii) To compensate simplifications in the underlying physical optics diffraction model for two-dimensional apertures [26] a size parameter dependent cross polarisation factor is implemented. It improves cross polarisation for diffraction and reflection by small particle facets. 2D patterns of P 11, –P 12/P 11 and P 22/P 11 and their azimuthal averages for slightly rough absorbing hexagonal prisms in fixed orientation are obtained and compared with results from the discrete dipole approximation. For particle orientations where shadowing is not negligible, improved phase functions are obtained by using a new method where the incident beam is divided into sub-beams with small triangular cross sections. The intersection points of the three sub-beam edges with the prism define the vertices of a triangle, which is treated by the beam tracer as an incidence-facing facet. This ensures that incident facing but shadowed crystal facets or regions thereof do not contribute to the phase functions. The method captures much of the fine detail contained in 2D scattering patterns obtained with DDA. This is important as speckle can be used for characterizing the size and roughness of small particles such as ice crystals.Peer reviewedFinal Accepted Versio