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In geometric topology, the **de Rham invariant** is a mod 2 invariant of a (4*k*+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]}^{[2]}

The de Rham invariant of a (4*k*+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.

**^**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