35 research outputs found

### 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