De Rham invariant explained
In geometric topology, the de Rham invariant is a mod 2 invariant of a (4k+1)-dimensional manifold, that is, an element of
– either 0 or 1. It can be thought of as the simply-connected
symmetric L-group
and thus analogous to the other invariants from L-theory: the
signature, a 4
k-dimensional invariant (either symmetric or quadratic,
), and the
Kervaire invariant, a (4
k+2)-dimensional
quadratic invariant
It is named for Swiss mathematician Georges de Rham, and used in surgery theory.[1]
Definition
The de Rham invariant of a (4k+1)-dimensional manifold can be defined in various equivalent ways:
- the rank of the 2-torsion in
as an integer mod 2;
;
where
is the Wu class of the normal bundle of
and
is the
Steenrod square; formally, as with all characteristic numbers, this is evaluated on the
fundamental class:
;
References
Notes and References
- John W. Morgan, A product formula for surgery obstructions, 1978
