Surgery obstruction

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In mathematics, specifically in surgery theory, the surgery obstructions define a map \theta \colon \mathcal{N} (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 necessarily a homomorphism) with the following property when n \geq 5:

A degree-one normal map (f,b) \colon M \to X is normally cobordant to a homotopy equivalence if and only if the image \theta (f,b)=0 in L_n (\mathbb{Z} [\pi_1 (X)]).

Sketch of the definition[edit]

The surgery obstruction of a degree-one normal map has a relatively complicated definition.

Consider a degree-one normal map (f,b) \colon M \to X. The idea in deciding the question whether it is normally cobordant to a homotopy equivalence is to try to systematically improve (f,b) so that the map f becomes m-connected (that means the homotopy groups \pi_* (f)=0 for * \leq m) for high m. It is a consequence of Poincaré duality that if we can achieve this for m > \lfloor n/2 \rfloor then the map f already is a homotopy equivalence. The word systematically above refers to the fact that one tries to do surgeries on M to kill elements of \pi_i (f). In fact it is more convenient to use homology of the universal covers to observe how connected the map f is. More precisely, one works with the surgery kernels K_i (\tilde M) : = \mathrm{ker} \{f_* \colon H_i (\tilde M) \rightarrow H_i (\tilde X)\} which one views as \mathbb{Z}[\pi_1 (X)]-modules. If all these vanish, then the map f is a homotopy equivalence. As a consequence of Poincaré duality on M and X there is a \mathbb{Z}[\pi_1 (X)]-modules Poincaré duality K^{n-i} (\tilde M) \cong K_i (\tilde M), so one only has to watch half of them, that means those for which i \leq \lfloor n/2 \rfloor.

Any degree-one normal map can be made \lfloor n/2 \rfloor-connected by the process called surgery below the middle dimension. This is the process of killing elements of K_i (\tilde M) for i < \lfloor n/2 \rfloor described here when we have p+q = n such that i = p < \lfloor n/2 \rfloor. After this is done there are two cases.

1. If n=2k then the only nontrivial homology group is the kernel K_k (\tilde M) : = \mathrm{ker} \{f_* \colon H_k (\tilde M) \rightarrow H_k (\tilde X)\}. It turns out that the cup-product pairings on M and X induce a cup-product pairing on K_k(\tilde M). This defines a symmetric bilinear form in case k=2l and a skew-symmetric bilinear form in case k=2l+1. It turns out that these forms can be refined to \varepsilon-quadratic forms, where \varepsilon = (-1)^k. These \varepsilon-quadratic forms define elements in the L-groups L_n (\pi_1 (X)).

2. If n=2k+1 the definition is more complicated. Instead of a quadratic form one obtains from the geometry a quadratic formation, which is a kind of automorphism of quadratic forms. Such a thing defines an element in the odd-dimensional L-group L_n (\pi_1 (X)).

If the element \theta (f,b) is zero in the L-group surgery can be done on M to modify f to a homotopy equivalence.

Geometrically the reason why this is not always possible is that performing surgery in the middle dimension to kill an element in K_k (\tilde M) possibly creates an element in K_{k-1} (\tilde M) when n = 2k or in K_{k} (\tilde M) when n=2k+1. So this possibly destroys what has already been achieved. However, if \theta (f,b) is zero, surgeries can be arranged in such a way that this does not happen.

Example[edit]

In the simply connected case the following happens.

If n = 2k+1 there is no obstruction.

If n = 4l then the surgery obstruction can be calculated as the difference of the signatures of M and X.

If n = 4l+2 then the surgery obstruction is the Arf-invariant of the associated kernel quadratic form over \mathbb{Z}_2.

References[edit]