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 – 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]

  1. ^ Morgan & Sullivan, The transversality characteristic class and linking cycles in surgery theory, 1974
  2. ^ John W. Morgan, A product formula for surgery obstructions, 1978
  3. ^ (Lusztig, Milnor & Peterson 1969)