In
surgery theory
In mathematics, specifically in geometric topology, surgery theory is a collection of techniques used to produce one finite-dimensional manifold from another in a 'controlled' way, introduced by . Milnor called this technique ''surgery'', while And ...
, a branch of
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, the stable normal bundle of a
differentiable manifold
In mathematics, a differentiable manifold (also differential manifold) is a type of manifold that is locally similar enough to a vector space to allow one to apply calculus. Any manifold can be described by a collection of charts (atlas). One ma ...
is an invariant which encodes the stable normal (dually, tangential) data. There are analogs for generalizations of manifold, notably
PL-manifolds and
topological manifold In topology, a branch of mathematics, a topological manifold is a topological space that locally resembles real ''n''-dimensional Euclidean space. Topological manifolds are an important class of topological spaces, with applications throughout mathe ...
s. There is also an analogue in
homotopy theory for
Poincaré space In algebraic topology, a Poincaré space is an ''n''-dimensional topological space with a distinguished element ''µ'' of its ''n''th homology group such that taking the cap product with an element of the ''k''th cohomology group yields an isomorphi ...
s, the Spivak spherical fibration, named after
Michael Spivak
Michael David Spivak (25 May 19401 October 2020)Biographical sketch in Notices of the AMS', Vol. 32, 1985, p. 576. was an American mathematician specializing in differential geometry, an expositor of mathematics, and the founder of Publish-or-Per ...
.
Construction via embeddings
Given an embedding of a manifold in
Euclidean space
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's Elements, Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics ther ...
(provided by the theorem of
Hassler Whitney
Hassler Whitney (March 23, 1907 – May 10, 1989) was an American mathematician. He was one of the founders of singularity theory, and did foundational work in manifolds, embeddings, immersions, characteristic classes, and geometric integratio ...
), it has a
normal bundle
In differential geometry, a field of mathematics, a normal bundle is a particular kind of vector bundle, complementary to the tangent bundle, and coming from an embedding (or immersion).
Definition
Riemannian manifold
Let (M,g) be a Riemannian m ...
. The embedding is not unique, but for high dimension of the Euclidean space it is unique up to
isotopy, thus the (class of the) bundle is unique, and called the ''stable normal bundle''.
This construction works for any
Poincaré space In algebraic topology, a Poincaré space is an ''n''-dimensional topological space with a distinguished element ''µ'' of its ''n''th homology group such that taking the cap product with an element of the ''k''th cohomology group yields an isomorphi ...
''X'': a finite
CW-complex
A CW complex (also called cellular complex or cell complex) is a kind of a topological space that is particularly important in algebraic topology. It was introduced by J. H. C. Whitehead (open access) to meet the needs of homotopy theory. This clas ...
admits a stably unique (up to homotopy) embedding in
Euclidean space
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's Elements, Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics ther ...
, via
general position
In algebraic geometry and computational geometry, general position is a notion of genericity for a set of points, or other geometric objects. It means the ''general case'' situation, as opposed to some more special or coincidental cases that ar ...
, and this embedding yields a spherical fibration over ''X''. For more restricted spaces (notably PL-manifolds and topological manifolds), one gets stronger data.
Details
Two embeddings
are ''isotopic'' if they are
homotopic
In topology, a branch of mathematics, two continuous functions from one topological space to another are called homotopic (from grc, ὁμός "same, similar" and "place") if one can be "continuously deformed" into the other, such a deforma ...
through embeddings. Given a manifold or other suitable space ''X,'' with two embeddings into Euclidean space
these will not in general be isotopic, or even maps into the same space (
need not equal
). However, one can embed these into a larger space
by letting the last
coordinates be 0:
:
.
This process of adjoining trivial copies of Euclidean space is called ''stabilization.''
One can thus arrange for any two embeddings into Euclidean space to map into the same Euclidean space (taking
), and, further, if
is sufficiently large, these embeddings are isotopic, which is a theorem.
Thus there is a unique stable isotopy class of embedding: it is not a particular embedding (as there are many embeddings), nor an isotopy class (as the target space is not fixed: it is just "a sufficiently large Euclidean space"), but rather a stable isotopy class of maps. The normal bundle associated with this (stable class of) embeddings is then the stable normal bundle.
One can replace this stable isotopy class with an actual isotopy class by fixing the target space, either by using
Hilbert space as the target space, or (for a fixed dimension of manifold
) using a fixed
sufficiently large, as ''N'' depends only on ''n'', not the manifold in question.
More abstractly, rather than stabilizing the embedding, one can take any embedding, and then take a vector bundle direct sum with a sufficient number of trivial line bundles; this corresponds exactly to the normal bundle of the stabilized embedding.
Construction via
classifying space
In mathematics, specifically in homotopy theory, a classifying space ''BG'' of a topological group ''G'' is the quotient of a weakly contractible space ''EG'' (i.e. a topological space all of whose homotopy groups are trivial) by a proper free ac ...
s
An ''n''-manifold ''M'' has a tangent bundle, which has a classifying map (up to homotopy)
:
Composing with the inclusion
yields (the homotopy class of a classifying map of) the stable tangent bundle. The normal bundle of an embedding
(
large) is an inverse
for
, such that the
Whitney sum
In mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in m ...
is trivial. The homotopy class of the composite
is independent of the choice of embedding, classifying the stable normal bundle
.
Motivation
There is no intrinsic notion of a normal vector to a manifold, unlike tangent or cotangent vectors – for instance, the normal space depends on which dimension one is embedding into – so the stable normal bundle instead provides a notion of a stable normal space: a normal space (and normal vectors) up to trivial summands.
Why stable normal, instead of stable tangent? Stable normal data is used instead of unstable tangential data because generalizations of manifolds have natural stable normal-type structures, coming from
tubular neighborhood
In mathematics, a tubular neighborhood of a submanifold of a smooth manifold is an open set around it resembling the normal bundle.
The idea behind a tubular neighborhood can be explained in a simple example. Consider a smooth curve in the pla ...
s and generalizations, but not unstable tangential ones, as the local structure is not smooth.
Spherical fibrations over a space ''X'' are classified by the homotopy classes of maps
to a
classifying space
In mathematics, specifically in homotopy theory, a classifying space ''BG'' of a topological group ''G'' is the quotient of a weakly contractible space ''EG'' (i.e. a topological space all of whose homotopy groups are trivial) by a proper free ac ...
, with
homotopy groups
In mathematics, homotopy groups are used in algebraic topology to classify topological spaces. The first and simplest homotopy group is the fundamental group, denoted \pi_1(X), which records information about loops in a space. Intuitively, homo ...
the
stable homotopy groups of spheres
:
.
The forgetful map
extends to a
fibration sequence
:
.
A
Poincaré space In algebraic topology, a Poincaré space is an ''n''-dimensional topological space with a distinguished element ''µ'' of its ''n''th homology group such that taking the cap product with an element of the ''k''th cohomology group yields an isomorphi ...
''X'' does not have a tangent bundle, but it does have a well-defined stable spherical
fibration, which for a differentiable manifold is the spherical fibration associated to the stable normal bundle; thus a primary obstruction to ''X'' having the homotopy type of a differentiable manifold is that the spherical fibration lifts to a vector bundle, i.e., the Spivak spherical fibration
must lift to
, which is equivalent to the map
being
null homotopic
In topology, a branch of mathematics, two continuous functions from one topological space to another are called homotopic (from grc, ὁμός "same, similar" and "place") if one can be "continuously deformed" into the other, such a deforma ...
Thus the bundle obstruction to the existence of a (smooth) manifold structure is the class
.
The secondary obstruction is the Wall
surgery obstruction In mathematics, specifically in surgery theory, the surgery obstructions define a map \theta \colon \mathcal (X) \to L_n (\pi_1 (X)) from the normal invariants to the L-groups which is in the first instance a set-theoretic map (that means not nece ...
.
Applications
The stable normal bundle is fundamental in
surgery theory
In mathematics, specifically in geometric topology, surgery theory is a collection of techniques used to produce one finite-dimensional manifold from another in a 'controlled' way, introduced by . Milnor called this technique ''surgery'', while And ...
as a primary obstruction:
*For a
Poincaré space In algebraic topology, a Poincaré space is an ''n''-dimensional topological space with a distinguished element ''µ'' of its ''n''th homology group such that taking the cap product with an element of the ''k''th cohomology group yields an isomorphi ...
''X'' to have the homotopy type of a smooth manifold, the map
must be
null homotopic
In topology, a branch of mathematics, two continuous functions from one topological space to another are called homotopic (from grc, ὁμός "same, similar" and "place") if one can be "continuously deformed" into the other, such a deforma ...
*For a homotopy equivalence
between two manifolds to be homotopic to a diffeomorphism, it must pull back the stable normal bundle on ''N'' to the stable normal bundle on ''M''.
More generally, its generalizations serve as replacements for the (unstable) tangent bundle.
References
{{Manifolds
Differential geometry
Surgery theory