Stable normal bundle

The stable normal bundle of a manifold is an important tool in differential topology , a branch of mathematics .

idea

According to Whitney's theorem , every manifold has an embedding in a Euclidean space, for which one can then consider the normal bundle. This embedding is not unambiguous in low codimensions , but unambiguous up to isotopic in sufficiently high codimensions , so that a normal bundle that is unambiguous except for isomorphism can be defined for embedding in high-dimensional Euclidean spaces.

definition

Let be a differentiable n-manifold with a tangent bundle . Be it ${\displaystyle M}$ ${\displaystyle TM}$

${\displaystyle \tau _{M}\colon M\to BO(n)}$

the classifying image of the tangential bundle. Here the Graßmann manifold denotes the classifying space for n-dimensional vector bundles. ${\displaystyle BO(n)}$

The normal bundle has a classifying map for embedding ${\displaystyle M\to \mathbb {R} ^{n+k}}$

${\displaystyle \nu _{M}\colon M\to BO(k)}$,

so the whitney sum

${\displaystyle \tau _{M}\oplus \nu _{M}\colon M\to BO(n+k)}$

is homotopic to a constant mapping .

Let it be the infinite-dimensional Graßmann manifold, the classifying space for stable vector bundles . It can be shown that the homotopy class of the composition does not depend on the embedding chosen. The stable vector bundle defined by this classifying mapping is called the stable normal bundle of . ${\displaystyle BO}$ ${\displaystyle \nu _{M}\colon M\to BO(k)\to BO}$ ${\displaystyle M}$

literature

Spivak, Michael: Spaces satisfying Poincaré duality. Topology 6 1967 77-101.