Surgery obstruction

Short description: Map from the normal invariants to the L-groups

In mathematics, specifically in surgery theory, the surgery obstructions define a map $\theta :{𝒩}(X)\to L_n({\pi }_1(X))$ 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 $n\ge 5$:

A degree-one normal map $(f,b):M\to X$ is normally cobordant to a homotopy equivalence if and only if the image $\theta (f,b)=0$ in $L_n(\mathbb{\mathbb{Z}}[{\pi }_1(X)])$.

Sketch of the definition

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

Consider a degree-one normal map $(f,b):M\to X$. 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 $*\le m$) 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 $K_i(\tilde{M}):=\mathrm{k}\mathrm{e}\mathrm{r}\{f_{*}:H_i(\tilde{M})\to H_i(\tilde{X})\}$ which one views as $\mathbb{\mathbb{Z}}[{\pi }_1(X)]$-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 $\mathbb{\mathbb{Z}}[{\pi }_1(X)]$-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 $i\le \lfloor n/2\rfloor$.

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 $K_k(\tilde{M}):=\mathrm{k}\mathrm{e}\mathrm{r}\{f_{*}:H_k(\tilde{M})\to H_k(\tilde{X})\}$. 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 $\epsilon$-quadratic forms, where $\epsilon =(-1{)}^k$. These $\epsilon$-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 ${\mathbb{\mathbb{Z}}}_2$.

References

• Browder, William (1972), Surgery on simply-connected manifolds, Berlin, New York: Springer-Verlag 
• 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, http://wwwmath.uni-muenster.de/u/lueck/publ/lueck/ictp.pdf 
• Ranicki, Andrew (2002), Algebraic and Geometric Surgery, Oxford Mathematical Monographs, Clarendon Press, ISBN 978-0-19-850924-0, http://www.maths.ed.ac.uk/~aar/books/index.htm 
• 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