Surgery obstruction

In mathematics, specifically in surgery theory, the surgery obstructions define a map ${\displaystyle \theta \colon {\mathcal {N}}(X)\to L_{n}(\pi _{1}(X))}$ 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 ${\displaystyle n\geq 5}$:

A degree-one normal map ${\displaystyle (f,b)\colon M\to X}$ is normally cobordant to a homotopy equivalence if and only if the image ${\displaystyle \theta (f,b)=0}$ in ${\displaystyle L_{n}(\mathbb {Z} [\pi _{1}(X)])}$.

Sketch of the definition

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

Consider a degree-one normal map ${\displaystyle (f,b)\colon M\to X}$. The idea in deciding the question whether it is normally cobordant to a homotopy equivalence is to try to systematically improve ${\displaystyle (f,b)}$ so that the map ${\displaystyle f}$ becomes ${\displaystyle m}$-connected (that means the homotopy groups ${\displaystyle \pi _{*}(f)=0}$ for ${\displaystyle *\leq m}$) for high ${\displaystyle m}$. It is a consequence of Poincaré duality that if we can achieve this for ${\displaystyle m>\lfloor n/2\rfloor }$ then the map ${\displaystyle f}$ already is a homotopy equivalence. The word systematically above refers to the fact that one tries to do surgeries on ${\displaystyle M}$ to kill elements of ${\displaystyle \pi _{i}(f)}$. In fact it is more convenient to use homology of the universal covers to observe how connected the map ${\displaystyle f}$ is. More precisely, one works with the surgery kernels${\displaystyle K_{i}({\tilde {M}}):=\mathrm {ker} \{f_{*}\colon H_{i}({\tilde {M}})\rightarrow H_{i}({\tilde {X}})\}}$ which one views as ${\displaystyle \mathbb {Z} [\pi _{1}(X)]}$-modules. If all these vanish, then the map ${\displaystyle f}$ is a homotopy equivalence. As a consequence of Poincaré duality on ${\displaystyle M}$ and ${\displaystyle X}$ there is a ${\displaystyle \mathbb {Z} [\pi _{1}(X)]}$-modules Poincaré duality ${\displaystyle K^{n-i}({\tilde {M}})\cong K_{i}({\tilde {M}})}$, so one only has to watch half of them, that means those for which ${\displaystyle i\leq \lfloor n/2\rfloor }$.

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

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

2. If ${\displaystyle n=2k+1}$ 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 ${\displaystyle L_{n}(\pi _{1}(X))}$.

If the element ${\displaystyle \theta (f,b)}$ is zero in the L-group surgery can be done on ${\displaystyle M}$ to modify ${\displaystyle f}$ 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 ${\displaystyle K_{k}({\tilde {M}})}$ possibly creates an element in ${\displaystyle K_{k-1}({\tilde {M}})}$ when ${\displaystyle n=2k}$ or in ${\displaystyle K_{k}({\tilde {M}})}$ when ${\displaystyle n=2k+1}$. So this possibly destroys what has already been achieved. However, if ${\displaystyle \theta (f,b)}$ 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 ${\displaystyle n=2k+1}$ there is no obstruction.

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

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

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 (PDF), 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