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A loop group extension of the odd Chern character
We show that the universal odd Chern form, defined on the stable unitary
group , extends to the loop group in a way that is closed with respect
to an equivariant-type differential. This provides an odd analogue to the
Bismut-Chern form. We also describe the associated transgression form, the
so-called Bismut-Chern-Simons form, and explicate some properties it inherits
as a differential form on the space of maps of a cylinder into the stable
unitary group. As a corollary, we obtain the Chern character homomorphism from
odd K-theory to the periodic cohomology of the free loop space, represented
geometrically on the level of differential forms.Comment: 12 pages, comments welcom
Pseudodeterminants and perfect square spanning tree counts
The pseudodeterminant of a square matrix is the last
nonzero coefficient in its characteristic polynomial; for a nonsingular matrix,
this is just the determinant. If is a symmetric or skew-symmetric
matrix then .
Whenever is the boundary map of a self-dual CW-complex ,
this linear-algebraic identity implies that the torsion-weighted generating
function for cellular -trees in is a perfect square. In the case that
is an \emph{antipodally} self-dual CW-sphere of odd dimension, the
pseudodeterminant of its th cellular boundary map can be interpreted
directly as a torsion-weighted generating function both for -trees and for
-trees, complementing the analogous result for even-dimensional spheres
given by the second author. The argument relies on the topological fact that
any self-dual even-dimensional CW-ball can be oriented so that its middle
boundary map is skew-symmetric.Comment: Final version; minor revisions. To appear in Journal of Combinatoric
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