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 .)
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 embedding (or more generally an immersion) 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 Whitney. 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.