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

The Spectral Geometry of the Mesh Matrices of Graphs

The mesh matrix $Mesh(G,T_0)$ of a connected finite graph $G=(V(G),E(G))=(vertices, edges) \ of \ G$ of with respect to a choice of a spanning tree $T_0 \subset G$ is defined and studied. It was introduced by Trent \cite{Trent1,Trent2}. Its characteristic polynomial $det(X \cdot Id -Mesh(G,T_0))$ is shown to equal $\Sigma_{j=0}^{N} \ (-1)^j \ ST_{j}(G,T_0)\ (X-1)^{N-j} \ (\star)$\ where$ST_j(G,T_0)$is the number of spanning trees of$G$meeting$E(G-T_0)$in j edges and$N=|E(G-T_0)|$. As a consequence, there are Tutte-type deletion-contraction formulae for computing this polynomial. Additionally,$Mesh(G,T_0) -Id$is of the special form$Y^t \cdot Y$; so the eigenvalues of the mesh matrix$Mesh(G,T_0)$are all real and are furthermore be shown to be$\ge +1$. It is shown that$Y \cdot Y^t$, called the mesh Laplacian, is a generalization of the standard graph Kirchhoff Laplacian$\Delta(H)= Deg -Adj$of a graph$H$.For example,$(\star)$generalizes the all minors matrix tree theorem for graphs$H$and gives a deletion-contraction formula for the characteristic polynomial of$\Delta(H)$. This generalization is explored in some detail. The smallest positive eigenvalue of the mesh Laplacian, a measure of flux, is estimated, thus extending the classical inequality for the Kirchoff Laplacian of graphs.Comment: 21 Page