Normal invariants
Encyclopedia
In mathematics, a normal map is a concept in geometric topology
due to William Browder
which is of fundamental importance in surgery theory
. Given a Poincaré complex
X, 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
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 due to C. T. C. Wall
to X actually being homotopy equivalent 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
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 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.
1. Given a Poincaré complex X (i.e. a CW-complex whose cellular chain complex satisfies Poincaré duality
) of formal dimension
, a normal map on X consists of
2. Given a Poincaré complex
(i.e. a CW-complex whose cellular chain complex satisfies Poincaré duality
) of formal dimension
, a normal map on 
(with respect to the tangent bundle) consists of
Two normal maps are equivalent if there exists a normal bordism between them.
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:
By Whitehead's theorem, the map
is a homotopy equivalence if and only if the surgery kernel is zero.
Notice that this new approach makes it necessary to classify the bordism classes of normal maps, which are the normal invariants. Contrarily to cobordism classes of maps, the normal invariants are a cohomology theory. Its coefficients are known in the case of topological manifolds. For the case of smooth manifolds, the coefficients of the theory are much more complicated.
. Recall that the main goal of surgery theory is to answer the questions:
1. Given a finite Poincaré complex
is there an 
-manifold homotopy equivalent to 
?
2. Given two homotopy equivalences
, where 
is there a diffeomorphism 
such that 
?
Notice that if the answer to these questions should be positive then it is a necessary condition that the answer to the following two questions is positive
1.' Given a finite Poincaré complex
is there a degree one normal map 
?
2.' Given two homotopy equivalences
, where 
is there a normal cobordism 
such that 
and 
?
This is of course an almost trivial observation, but it is important because it turns out that there is an effective theory which answers question 1.' and also an effective theory which answers question 1. provided the answer to 1.' is yes. Similarly for questions 2. and 2.' Notice also that we can phrase the questions as follows:
1.' Is
?
2.' Is
in 
?
Hence studying
is really a first step in trying to understand the surgery structure set 
which is the main goal in surgery theory. The point is that 
is much more accessible from the point of view of algebraic topology as is explained below.
with normal bundles. Recall that a (smooth) manifold has a unique tangent bundle and a unique stable normal bundle. But a finite Poincaré complex does not possess such a unique bundle. Nevertheless, it possesses a substitute - a unique in some sense spherical fibration - the so called Spivak normal fibration. This has a property that if 
is homotopy equivalent to a manifold then the spherical fibration associated to the pullback of the normal bundle of that manifold is isomorphic to the Spivak normal fibration. So it follows that if 
then the Spivak normal fibration has a bundle reduction. By the Pontrjagin-Thom construction the converse is also true.
This can be formulated in terms of homotopy theory. Recall
the classifying space for stable spherical fibrations, 
the classifying space for stable vector bundles and the map 
which is induced by the inclusion 
and which corresponds to taking the associated spherical fibration of a vector bundle. In fact we have a fibration sequence 
. The Spivak normal fibration is classified by a map 
. It has a vector bundle reduction if and only if 
has a lift 
. This is equivalent to requiring that the composition 
is null-homotopic.
Note that the homotopy groups of
are known in certain low-dimensions and are non-trivial which suggests the possibility that the above condition can fail for some 
. There are in fact such finite Poincaré complexes, and the first example was obtained by Gitler and Stasheff, yielding thus an example of a Poincaré complex not homotopy equivalent to a manifold.
2.' Relativizing the above considerations one obtains an (unnatural) bijection
a structure of an abelian group since the space 
is a loop space and in fact an infinite loop space so the normal invariants are a zeroth cohomology group of an extraordinary cohomology theory defined by that inifinite loop space. Note that similar ideas apply in the other categories of manifolds and one has bijections
It is well known that the spaces
are mutually not homotopy equivalent and hence one obtains three different cohomology theories.
Sullivan analyzed the cases
and 
. He showed that these spaces possess alternative inifinite loop space structures which are in fact better from the following point of view: Recall that there is a surgery obstruction map from normal invariants to the L-group. With the above described groups structure on the normal invariants this map is NOT a homomorphism. However, with the group structure from Sullivan's theorem it becomes a homomorphism in the categories 
, and 
. His theorem also links these new group structures to the well-known cohomology theories: the singular cohomology and real K-theory.
Geometric topology
In mathematics, geometric topology is the study of manifolds and maps between them, particularly embeddings of one manifold into another.- Topics :...
due to William Browder
William Browder (mathematician)
William Browder is an American mathematician, specializing in algebraic topology, differential topology and differential geometry...
which is of fundamental importance in surgery theory
Surgery theory
In mathematics, specifically in geometric topology, surgery theory is a collection of techniques used to produce one manifold from another in a 'controlled' way, introduced by . Surgery refers to cutting out parts of the manifold and replacing it with a part of another manifold, matching up along...
. Given a Poincaré complex
Poincaré complex
In mathematics, and especially topology, a Poincaré complex is a generalisation of a manifold.The homology groups of a closed, orientable manifold are related in certain ways. This relationship came to be known as the Poincaré duality. Poincaré duality gives an isomorphism between homology and...
X, 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
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 data. There are analogs for generalizations of manifold, notably PL-manifolds and topological manifolds...
and a Thom
René Thom
René Frédéric Thom was a French mathematician. He made his reputation as a topologist, moving on to aspects of what would be called singularity theory; he became world-famous among the wider academic community and the educated general public for one aspect of this latter interest, his work as...
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 due to C. T. C. WallC. T. C. Wall
Charles Terence Clegg Wall is a leading British mathematician, educated at Marlborough and Trinity College, Cambridge. He is an emeritus professor of the University of Liverpool, where he was first appointed Professor in 1965...
to X actually being homotopy equivalent 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
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 of a manifold. Two manifolds are cobordant if their disjoint union is the boundary of a manifold one dimension higher. The name comes...
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 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 whose cellular chain complex satisfies Poincaré duality
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...
) 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 bundleStable normal bundleIn surgery theory, a branch of mathematics, the stable normal bundle of a differentiable manifold is an invariant which encodes the stable normal data. There are analogs for generalizations of manifold, notably PL-manifolds and topological manifolds...
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 whose cellular chain complex satisfies Poincaré dualityPoincaré 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...
) 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 bundleTangent bundleIn differential geometry, the tangent bundle of a differentiable manifold M is the disjoint unionThe disjoint union assures that for any two points x1 and x2 of manifold M the tangent spaces T1 and T2 have no common vector...
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.
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 ThomThom
The surname Thom is of Scottish origin, from the city of Aberdeen, Aberdeenshire and Angus, is a sept of the Clan MacThomas.Thom is also a first name variant of the abbreviation "Tom" of "Thomas" that holds the "h".-People With Thom as Surname:...
: 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 
.)
By Whitehead's theorem, the map
is a homotopy equivalence if and only if the surgery kernel is zero.
- The bundle data implies the following: Suppose that an element
(the relative homotopy group of 
) can be represented by an embeddingEmbeddingIn mathematics, an embedding is one instance of some mathematical structure contained within another instance, such as a group that is a subgroup....
(or more generally an immersionImmersionImmersion may refer to:* Immersion therapy overcoming fears through confrontation* Baptism by immersion* Immersion Games a developer of video games...
) with a null-homotopy of
. Then it can be represented by an embedding (or immersion) whose normal bundle is stably trivial. This observation is important since surgery is only possible on embeddings with a trivial normal bundle. For example, if 
is less than half the dimension of 
, every map 
is homotopic to an embedding by a theorem of WhitneyWhitney-Places:Canada* Whitney, OntarioUnited Kingdom* Whitney-on-Wye, Herefordshire* Witney, OxfordshireUnited States* Whitney, California, a community in Placer County* Whitney, a neighborhood in Beverly, Visalia, California...
. On the other hand, every stably trivial normal bundle of such an embedding is automatically trivial, since
for 
. Therefore, surgery on normal maps can always be done below the middle dimension. This is not true for arbitrary maps.
Notice that this new approach makes it necessary to classify the bordism classes of normal maps, which are the normal invariants. Contrarily to cobordism classes of maps, the normal invariants are a cohomology theory. Its coefficients are known in the case of topological manifolds. For the case of smooth manifolds, the coefficients of the theory are much more complicated.
Normal invariants versus structure set
There are two reasons why it is important to study the set. Recall that the main goal of surgery theory is to answer the questions:1. Given a finite Poincaré complex
is there an -manifold homotopy equivalent to ?2. Given two homotopy equivalences
, where is there a diffeomorphism such that ?Notice that if the answer to these questions should be positive then it is a necessary condition that the answer to the following two questions is positive
1.' Given a finite Poincaré complex
is there a degree one normal map ?2.' Given two homotopy equivalences
, where is there a normal cobordism such that and ?This is of course an almost trivial observation, but it is important because it turns out that there is an effective theory which answers question 1.' and also an effective theory which answers question 1. provided the answer to 1.' is yes. Similarly for questions 2. and 2.' Notice also that we can phrase the questions as follows:
1.' Is
?2.' Is
in ?Hence studying
is really a first step in trying to understand the surgery structure set which is the main goal in surgery theory. The point is that is much more accessible from the point of view of algebraic topology as is explained below.Homotopy theory
1.' Let X be a finite n-dimensional Poincaré complex. It is useful to use the definition of with normal bundles. Recall that a (smooth) manifold has a unique tangent bundle and a unique stable normal bundle. But a finite Poincaré complex does not possess such a unique bundle. Nevertheless, it possesses a substitute - a unique in some sense spherical fibration - the so called Spivak normal fibration. This has a property that if is homotopy equivalent to a manifold then the spherical fibration associated to the pullback of the normal bundle of that manifold is isomorphic to the Spivak normal fibration. So it follows that if then the Spivak normal fibration has a bundle reduction. By the Pontrjagin-Thom construction the converse is also true.This can be formulated in terms of homotopy theory. Recall
the classifying space for stable spherical fibrations, the classifying space for stable vector bundles and the map which is induced by the inclusion and which corresponds to taking the associated spherical fibration of a vector bundle. In fact we have a fibration sequence . The Spivak normal fibration is classified by a map . It has a vector bundle reduction if and only if has a lift . This is equivalent to requiring that the composition is null-homotopic.Note that the homotopy groups of
are known in certain low-dimensions and are non-trivial which suggests the possibility that the above condition can fail for some . There are in fact such finite Poincaré complexes, and the first example was obtained by Gitler and Stasheff, yielding thus an example of a Poincaré complex not homotopy equivalent to a manifold.2.' Relativizing the above considerations one obtains an (unnatural) bijection
Different categories
The above bijection gives a structure of an abelian group since the space is a loop space and in fact an infinite loop space so the normal invariants are a zeroth cohomology group of an extraordinary cohomology theory defined by that inifinite loop space. Note that similar ideas apply in the other categories of manifolds and one has bijections-
, and 
, and 
It is well known that the spaces
-
, 
and 
are mutually not homotopy equivalent and hence one obtains three different cohomology theories.
Sullivan analyzed the cases
and . He showed that these spaces possess alternative inifinite loop space structures which are in fact better from the following point of view: Recall that there is a surgery obstruction map from normal invariants to the L-group. With the above described groups structure on the normal invariants this map is NOT a homomorphism. However, with the group structure from Sullivan's theorem it becomes a homomorphism in the categories , and . His theorem also links these new group structures to the well-known cohomology theories: the singular cohomology and real K-theory.