de Rham invariant

{{Short description|Mod 2 invariant of (4k+1)-dimensional manifold}}

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

It is named for Swiss mathematician Georges de Rham, and used in surgery theory.{{citation|last1=Morgan|first1= John W|author1-link=John Morgan (mathematician)| last2= Sullivan|first2= Dennis P.|author2-link=Dennis Sullivan|

title=The transversality characteristic class and linking cycles in surgery theory|

journal=Annals of Mathematics| series=2|volume= 99 |year=1974|issue= 3|pages= 463–544|mr=0350748 |doi=10.2307/1971060|jstor= 1971060}}John W. Morgan, [https://books.google.com/books?id=PN7QbZi1gdgC A product formula for surgery obstructions], 1978

Definition

The de Rham invariant of a (4k+1)-dimensional manifold can be defined in various equivalent ways:{{Harv|Lusztig|Milnor|Peterson|1969}}

the rank of the 2-torsion in $H_{2k}(M),$ as an integer mod 2;
the Stiefel–Whitney number $w_2w_{4k-1}$ ;
the (squared) Wu number, $v_{2k}Sq^1v_{2k},$ where $v_{2k} \in H^{2k}(M;Z_2)$ is the Wu class of the normal bundle of $M$ and $Sq^1$ is the Steenrod square; formally, as with all characteristic numbers, this is evaluated on the fundamental class: $(v_{2k}Sq^1v_{2k},[M])$ ;
in terms of a semicharacteristic.

References

{{citation

| first1 = George | last1 = Lusztig | authorlink1 = George Lusztig

| first2 = John | last2 = Milnor | authorlink2 = John Milnor

| first3 = Franklin P. | last3 = Peterson |authorlink3 = Franklin P. Peterson

| title = Semi-characteristics and cobordism

| doi-access = free }}

Chess, Daniel, [http://projecteuclid.org/DPubS?service=UI&version=1.0&verb=Display&handle=euclid.bams/1183547688 A Poincaré-Hopf type theorem for the de Rham invariant], 1980

Category:Geometric topology

Category:Surgery theory