“De Rham invariant”的意思、由来-开放百科全书

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De Rham invariant

释义

Definition
References

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

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

1. ^{{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|pages= 463–544|mr=0350748 |doi=10.2307/1971060}}
2. ^John W. Morgan, [https://books.google.com/books?id=PN7QbZi1gdgC A product formula for surgery obstructions], 1978
3. ^{{Harv|Lusztig|Milnor|Peterson|1969}}

{{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
| journal = Topology | volume = 8 | year = 1969 |pages = 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

2 : Geometric topology|Surgery theory

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