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.
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.
Depending on the category of manifolds (differentiable, piecewise-linear, or topological), there are similarly defined, but inequivalent, concepts of normal maps and normal invariants.
Surgery on normal maps allows one to systematically kill elements in the relative homotopy groups by representing them as embeddings with trivial normal bundle.
Hence it is possible to switch between the definitions which turns out to be quite convenient.
(i.e. a CW-complex whose cellular chain complex satisfies Poincaré duality) of formal dimension
Therefore, in the classical surgery approach to our question, one starts with a normal map
Contrarily to cobordism classes of maps, the normal invariants are a cohomology theory.
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.
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.'
is really a first step in trying to understand the surgery structure set
is much more accessible from the point of view of algebraic topology as is explained below.
But a finite Poincaré complex does not possess such a unique bundle.
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.
then the Spivak normal fibration has a bundle reduction.
the classifying space for stable spherical fibrations,
the classifying space for stable vector bundles and the map
and which corresponds to taking the associated spherical fibration of a vector bundle.
The Spivak normal fibration is classified by a map
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,[citation needed] yielding thus an example of a Poincaré complex not homotopy equivalent to a manifold.
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.
He showed that these spaces possess alternative infinite 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