35 research outputs found

    Unoriented geometric functors

    Full text link
    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

    Full text link
    The mesh matrix Mesh(G,T0)Mesh(G,T_0) of a connected finite graph G=(V(G),E(G))=(vertices,edges)Β ofΒ GG=(V(G),E(G))=(vertices, edges) \ of \ G of with respect to a choice of a spanning tree T0βŠ‚GT_0 \subset G is defined and studied. It was introduced by Trent \cite{Trent1,Trent2}. Its characteristic polynomial det(Xβ‹…Idβˆ’Mesh(G,T0))det(X \cdot Id -Mesh(G,T_0)) is shown to equal $\Sigma_{j=0}^{N} \ (-1)^j \ ST_{j}(G,T_0)\ (X-1)^{N-j} \ (\star)Β where \ where ST_j(G,T_0)isthenumberofspanningtreesof is the number of spanning trees of Gmeeting meeting E(G-T_0)injedgesand in j edges and N=|E(G-T_0)|.Asaconsequence,thereareTutteβˆ’typedeletionβˆ’contractionformulaeforcomputingthispolynomial.Additionally,. As a consequence, there are Tutte-type deletion-contraction formulae for computing this polynomial. Additionally, Mesh(G,T_0) -Idisofthespecialform is of the special form Y^t \cdot Y;sotheeigenvaluesofthemeshmatrix; so the eigenvalues of the mesh matrix Mesh(G,T_0)areallrealandarefurthermorebeshowntobe are all real and are furthermore be shown to be \ge +1.Itisshownthat. It is shown that Y \cdot Y^t,calledthemeshLaplacian,isageneralizationofthestandardgraphKirchhoffLaplacian, called the mesh Laplacian, is a generalization of the standard graph Kirchhoff Laplacian \Delta(H)= Deg -Adjofagraph of a graph H.Forexample,.For example, (\star)generalizestheallminorsmatrixtreetheoremforgraphs generalizes the all minors matrix tree theorem for graphs Handgivesadeletionβˆ’contractionformulaforthecharacteristicpolynomialof and gives a deletion-contraction formula for the characteristic polynomial of \Delta(H)$. 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