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 $\mathbb{Z}_2$ .

References

Browder, William (1972), Surgery on simply-connected manifolds, Berlin, New York: Springer-Verlag, MR 0358813
Lück, Wolfgang (2002), A basic introduction to surgery theory, ICTP Lecture Notes Series 9, Band 1, of the school "High-dimensional manifold theory" in Trieste, May/June 2001, Abdus Salam International Centre for Theoretical Physics, Trieste 1-224
Ranicki, Andrew (2002), Algebraic and Geometric Surgery, Oxford Mathematical Monographs, Clarendon Press, ISBN 978-0-19-850924-0, MR 2061749
Wall, C. T. C. (1999), Surgery on compact manifolds, Mathematical Surveys and Monographs 69 (2nd ed.), Providence, R.I.: American Mathematical Society, ISBN 978-0-8218-0942-6, MR 1687388