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For $A$ an abelian group and given a symmetric bilinear form
then a characteristic element of the bilinear form is an element $\lambda\in A$ such that for all $x\in A$ the equation
holds, or in other words such that
is divisible by $2\in \mathbb{Z}$.
Often this is considered for the bilinear form being an intersection product in which case $A$ is a cohomology group and so in this case one also often speaks of characteristic cohomology classes.
A characteristic element determines a quadratic refinement $q_\lambda$ of $\langle-,-\rangle$ by
This follows by trivial direct computation:
the only point being that the prefactor $\tfrac{1}{2}$ indeed makes sense, by def. .
Characteristic elements for the intersection pairing on integral ordinary cohomology are integral lifts of the Wu classes, hence integral Wu structures.
Discussion of the refinement of integral Wu structures as the characteristic elements of the intersection product from ordinary cohomology to ordinary differential cohomology is in
Further discussion of the use of characteristic elements in the construction of Theta characteristics is in
Last revised on August 23, 2020 at 00:50:53. See the history of this page for a list of all contributions to it.