Stiefel-Whitney class
A characteristic class with values in , defined for real vector bundles. The Stiefel–Whitney classes are denoted by
,
, and for a real vector bundle
over a topological space
, the class
lies in
. These classes were introduced by E. Stiefel [1] and H. Whitney [2] and have the following properties. 1) For two real vector bundles
over a common base,
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in other words, , where
is the complete Stiefel–Whitney class. 2) For the one-dimensional universal bundle
over
the equality
holds, where
is the non-zero element of the group
. These two properties together with naturality for induced bundles define the Stiefel–Whitney classes uniquely. The Stiefel–Whitney classes are stable, i.e.
, where
is the trivial bundle, and
for
. For an oriented
-dimensional vector bundle
over a base
,
coincides with the reduction modulo 2 of the Euler class.
For a vector bundle over
, let
be the Thom space of this bundle. Further, let
be the Thom isomorphism. Then the complete Stiefel–Whitney class
coincides with
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where is the complete Steenrod square. This property of Stiefel–Whitney classes can be used as their definition. Stiefel–Whitney classes are homotopy invariants in the sense that they coincide for fibre-wise homotopically-equivalent bundles over a common base.
Any characteristic class with values in , defined for real vector bundles, can be expressed by Stiefel–Whitney classes: The rings
and
are rings of formal power series in the Stiefel–Whitney classes:
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References
[1] | E. Stiefel, "Richtungsfelden und Fernparallelismus in ![]() |
[2] | H. Whitney, "Topological properties of differentiable manifolds" Bull. Amer. Math. Soc. , 43 (1937) pp. 785–805 |
[3] | J.W. Milnor, J.D. Stasheff, "Characteristic classes" , Princeton Univ. Press (1974) |
[4] | R.E. Stong, "Notes on cobordism theory" , Princeton Univ. Press (1968) |
[5] | N.E. Steenrod, "The topology of fibre bundles" , Princeton Univ. Press (1951) |
Comments
The notation denotes the product of the Abelian groups
, while
is the direct sum; the notation
means that there is a graded homomorphism of degree
:
. For the classifying spaces
and
see Classifying space.
References
[a1] | D. Husemoller, "Fibre bundles" , McGraw-Hill (1966) |
Stiefel-Whitney class. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Stiefel-Whitney_class&oldid=12435