De Rham invariant
Appearance
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 4k-dimensional invariant (either symmetric or quadratic, ), and the Kervaire invariant, a (4k+2)-dimensional quadratic invariant
It is named for Swiss mathematician Georges de Rham, and used in surgery theory.[1][2]
Definition
[edit]The de Rham invariant of a (4k+1)-dimensional manifold can be defined in various equivalent ways:[3]
- the rank of the 2-torsion in as an integer mod 2;
- the Stiefel–Whitney number ;
- the (squared) Wu number, 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: ;
- in terms of a semicharacteristic.
References
[edit]- ^ Morgan, John W; Sullivan, Dennis P. (1974), "The transversality characteristic class and linking cycles in surgery theory", Annals of Mathematics, 2, 99 (3): 463–544, doi:10.2307/1971060, JSTOR 1971060, MR 0350748
- ^ John W. Morgan, A product formula for surgery obstructions, 1978
- ^ (Lusztig, Milnor & Peterson 1969)
- Lusztig, George; Milnor, John; Peterson, Franklin P. (1969), "Semi-characteristics and cobordism", Topology, 8 (4): 357–360, doi:10.1016/0040-9383(69)90021-4, MR 0246308
- Chess, Daniel, A Poincaré-Hopf type theorem for the de Rham invariant, 1980