In geometric topology

Geometric topology

In mathematics, geometric topology is the study of manifolds and maps between them, particularly embeddings of one manifold into another.- Topics :...

, 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

L-theory

Algebraic L-theory is the K-theory of quadratic forms; the term was coined by C. T. C. Wall,with L being used as the letter after K. Algebraic L-theory, also known as 'hermitian K-theory',is important in surgery theory.-Definition:...

  and thus analogous to the other invariants from L-theory: the signature

Signature (topology)

In the mathematical field of topology, the signature is an integer invariant which is defined for an oriented manifold M of dimension d=4k divisible by four ....

, a 4k-dimensional invariant, and the Kervaire invariant, a (4k+2)-dimensional invariant.

It is named for Swiss mathematician Georges de Rham

Georges de Rham

Georges de Rham was a Swiss mathematician, known for his contributions to differential topology.He studied at the University of Lausanne and then in Paris for a doctorate, becoming a lecturer in Lausanne in 1931; where he held positions until retirement in 1971; he held positions in Geneva in...

, and used in surgery theory

Surgery theory

In mathematics, specifically in geometric topology, surgery theory is a collection of techniques used to produce one manifold from another in a 'controlled' way, introduced by . Surgery refers to cutting out parts of the manifold and replacing it with a part of another manifold, matching up along...

.

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;
• the Stiefel–Whitney number ;
• the (squared) Wu number, where is the Wu class and is the Steenrod square; formally, as with all characteristic numbers, this is evaluated on the fundamental class
  Fundamental class
  In mathematics, the fundamental class is a homology class [M] associated to an oriented manifold M, which corresponds to "the whole manifold", and pairing with which corresponds to "integrating over the manifold"...
  
  : ;
• in terms of a semicharacteristic.