Neha Patil (Editor)

De Rham invariant

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In geometric topology, the de Rham invariant is a mod 2 invariant of a (4k+1)-dimensional manifold, that is, an element of Z / 2 – either 0 or 1. It can be thought of as the simply-connected symmetric L-group L 4 k + 1 , and thus analogous to the other invariants from L-theory: the signature, a 4k-dimensional invariant (either symmetric or quadratic, L 4 k L 4 k ), and the Kervaire invariant, a (4k+2)-dimensional quadratic invariant L 4 k + 2 .

It is named for Swiss mathematician Georges de Rham, and used in surgery theory.

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 H 2 k ( M ) , as an integer mod 2;
  • the Stiefel–Whitney number w 2 w 4 k 1 ;
  • the (squared) Wu number, v 2 k S q 1 v 2 k , where v 2 k H 2 k ( M ; Z 2 ) is the Wu class of the normal bundle of M and S q 1 is the Steenrod square ; formally, as with all characteristic numbers, this is evaluated on the fundamental class: ( v 2 k S q 1 v 2 k , [ M ] ) ;
  • in terms of a semicharacteristic.
  • References

    De Rham invariant Wikipedia