In mathematics, a normal map is a concept in
geometric topology
In mathematics, geometric topology is the study of manifolds and maps between them, particularly embeddings of one manifold into another.
History
Geometric topology as an area distinct from algebraic topology may be said to have originated i ...
due to
William Browder which is of fundamental importance 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 An ...
. Given a
Poincaré complex In mathematics, and especially topology, a Poincaré complex (named after the mathematician Henri Poincaré) is an abstraction of the singular chain complex of a closed, orientable manifold.
The singular homology and cohomology groups of a close ...
''X'' (more geometrically 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 ...
), a normal map on ''X'' endows the space, roughly speaking, with some of the homotopy-theoretic global structure of a closed manifold. In particular, ''X'' has a good candidate for a
stable normal bundle
In surgery theory, a branch of mathematics, the stable normal bundle of a differentiable manifold is an invariant which encodes the stable normal (dually, tangential) data. There are analogs for generalizations of manifold, notably PL-manifolds a ...
and a
Thom collapse map, which is equivalent to there being a map from a manifold ''M'' to ''X'' matching the fundamental classes and preserving normal bundle information. If the dimension of ''X'' is
5 there is then only the algebraic topology
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 neces ...
due to
C. T. C. Wall
Charles Terence Clegg "Terry" Wall (born 14 December 1936) is a British mathematician, educated at Marlborough College, Marlborough and Trinity College, Cambridge. He is an :wikt:emeritus, emeritus professor of the University of Liverpool, where ...
to ''X'' actually being
homotopy equivalent
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 ...
to a closed manifold. Normal maps also apply to the study of the uniqueness of manifold structures within a homotopy type, which was pioneered by
Sergei Novikov.
The
cobordism
In mathematics, cobordism is a fundamental equivalence relation on the class of compact manifolds of the same dimension, set up using the concept of the boundary (French '' bord'', giving ''cobordism'') of a manifold. Two manifolds of the same dim ...
classes of normal maps on ''X'' are called normal invariants. Depending on the category of manifolds (differentiable, piecewise-linear, or topological), there are similarly defined, but inequivalent, concepts of normal maps and normal invariants.
It is possible to perform
surgery
Surgery ''cheirourgikē'' (composed of χείρ, "hand", and ἔργον, "work"), via la, chirurgiae, meaning "hand work". is a medical specialty that uses operative manual and instrumental techniques on a person to investigate or treat a pat ...
on normal maps, meaning surgery on the domain manifold, and preserving the map. Surgery on normal maps allows one to systematically kill elements in the relative homotopy groups by representing them as embeddings ''with trivial normal bundle''.
Definition
There are two equivalent definitions of normal maps, depending on whether one uses normal bundles or tangent bundles of manifolds. Hence it is possible to switch between the definitions which turns out to be quite convenient.
1. Given a Poincaré complex ''X'' (i.e. a
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 cla ...
whose cellular chain complex satisfies
Poincaré duality
In mathematics, the Poincaré duality theorem, named after Henri Poincaré, is a basic result on the structure of the homology and cohomology groups of manifolds. It states that if ''M'' is an ''n''-dimensional oriented closed manifold (compact a ...
) of formal dimension
, a normal map on ''X'' consists of
* a map
from some closed ''n''-dimensional manifold ''M'',
* a bundle
over ''X'', and a stable map from the
stable normal bundle
In surgery theory, a branch of mathematics, the stable normal bundle of a differentiable manifold is an invariant which encodes the stable normal (dually, tangential) data. There are analogs for generalizations of manifold, notably PL-manifolds a ...
of
to
, and
* usually the normal map is supposed to be of degree one. That means that the fundamental class of
should be mapped under
to the fundamental class of
:
.
2. Given a Poincaré complex
(i.e. a
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 cla ...
whose cellular chain complex satisfies
Poincaré duality
In mathematics, the Poincaré duality theorem, named after Henri Poincaré, is a basic result on the structure of the homology and cohomology groups of manifolds. It states that if ''M'' is an ''n''-dimensional oriented closed manifold (compact a ...
) of formal dimension
, a normal map on
(with respect to the tangent bundle) consists of
* a map
from some closed
-dimensional manifold
,
* a bundle
over
, and a stable map from the stable
tangent bundle
In differential geometry, the tangent bundle of a differentiable manifold M is a manifold TM which assembles all the tangent vectors in M . As a set, it is given by the disjoint unionThe disjoint union ensures that for any two points and of ...
of
to
, and
* similarly as above it is required that the fundamental class of
should be mapped under
to the fundamental class of
:
.
Two normal maps are equivalent if there exists a normal bordism between them.
Role in surgery theory
Surgery on maps versus surgery on normal maps
Consider the question:
: Is the Poincaré complex ''X'' of formal dimension ''n'' homotopy-equivalent to a closed ''n''-manifold?
A naive surgery approach to this question would be: start with some map
from some manifold
to
, and try to do surgery on it to make a homotopy equivalence out of it. Notice the following: Since our starting map was arbitrarily chosen, and surgery always produces cobordant maps, this procedure has to be performed (in the worst case) for all cobordism classes of maps
. This kind of cobordism theory is a homology theory whose coefficients have been calculated by
Thom: therefore the cobordism classes of such maps are computable at least in theory for all spaces
.
However, it turns out that it is very difficult to decide whether it is possible to make a homotopy equivalence out of the map by means of surgery, whereas the same question is much easier when the map comes with the extra structure of a normal map. Therefore, in the classical surgery approach to our question, one starts with a normal map
(suppose there exists any), and performs surgery on it. This has several advantages:
* The map being of degree one implies that the homology of
splits as a direct sum of the homology of
and the so-called surgery kernel
, that is
. (Here we suppose that
induces an isomorphism of fundamental groups and use homology with local coefficients in