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 (more geometrically a Poincaré space), 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 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.

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.

Given a Poincaré complex X (i.e. a CW-complex whose cellular chain complex satisfies Poincaré duality, a normal map on X consists of

• from some closed n-dimensional manifold M,

• over X, and a stable map from the stable normal bundle, and

• usually the normal map is supposed to be of degree one. That means that the fundamental class.

(with respect to the tangent bundle) consists of

• ,

• , and a stable map from the stable tangent bundle, and

• .

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:

1. Is the Poincaré complex X of formal dimension n homotopy-equivalent to a closed n-manifold?

. This kind of cobordism theory is a homology theory whose coefficients have been calculated by Thom.

(suppose there exists any), and performs surgery on it. This has several advantages:

• and the so-called surgery kernel.)

By Whitehead's theorem is a homotopy equivalence if and only if the surgery kernel is zero.

• ) can be represented by an embedding (or more generally an immersion is homotopic to an embedding by a theorem of Whitney. 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

. Recall that the main goal of surgery theory is to answer the questions:

?

?

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

?

?

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:

?

?

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 then the Spivak normal fibration has a bundle reduction. By the Pontrjagin-Thom construction the converse is also true.

is null-homotopic.

. 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

a structure of an abelian group 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 infinite 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. His theorem also links these new group structures to the well-known cohomology theories: the singular cohomology and real K-theory.

References