Surgery obstruction

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
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
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 $$\mathbb{Z}_2$$.