In mathematics, specifically in surgery theory, the surgery obstructions define a map θ : N ( X ) → L n ( π 1 ( X ) ) {\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 n ≥ 5 {\displaystyle n\geq 5}:

A degree-one normal map ( f , b ) : M → X {\displaystyle (f,b)\colon M\to X} is normally cobordant to a homotopy equivalence if and only if the image θ ( f , b ) = 0 {\displaystyle \theta (f,b)=0} in L n ( Z [ π 1 ( X ) ] ) {\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 ( f , b ) : M → X {\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 ( f , b ) {\displaystyle (f,b)} so that the map f {\displaystyle f} becomes m {\displaystyle m}-connected (that means the homotopy groups π ∗ ( f ) = 0 {\displaystyle \pi _{*}(f)=0} for ∗ ≤ m {\displaystyle *\leq m}) for high m {\displaystyle m}. It is a consequence of Poincaré duality that if we can achieve this for m > ⌊ n / 2 ⌋ {\displaystyle m>\lfloor n/2\rfloor } then the map f {\displaystyle f} already is a homotopy equivalence. The word systematically above refers to the fact that one tries to do surgeries on M {\displaystyle M} to kill elements of π i ( f ) {\displaystyle \pi _{i}(f)}. In fact it is more convenient to use homology of the universal covers to observe how connected the map f {\displaystyle f} is. More precisely, one works with the surgery kernels K i ( M ~ ) := k e r { f ∗ : H i ( M ~ ) → H i ( X ~ ) } {\displaystyle K_{i}({\tilde {M}}):=\mathrm {ker} \{f_{*}\colon H_{i}({\tilde {M}})\rightarrow H_{i}({\tilde {X}})\}} which one views as Z [ π 1 ( X ) ] {\displaystyle \mathbb {Z} [\pi _{1}(X)]}-modules. If all these vanish, then the map f {\displaystyle f} is a homotopy equivalence. As a consequence of Poincaré duality on M {\displaystyle M} and X {\displaystyle X} there is a Z [ π 1 ( X ) ] {\displaystyle \mathbb {Z} [\pi _{1}(X)]}-modules Poincaré duality K n − i ( M ~ ) ≅ K i ( M ~ ) {\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 i ≤ ⌊ n / 2 ⌋ {\displaystyle i\leq \lfloor n/2\rfloor }.

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

1. If n = 2 k {\displaystyle n=2k} then the only nontrivial homology group is the kernel K k ( M ~ ) := k e r { f ∗ : H k ( M ~ ) → H k ( X ~ ) } {\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 M {\displaystyle M} and X {\displaystyle X} induce a cup-product pairing on K k ( M ~ ) {\displaystyle K_{k}({\tilde {M}})}. This defines a symmetric bilinear form in case k = 2 l {\displaystyle k=2l} and a skew-symmetric bilinear form in case k = 2 l + 1 {\displaystyle k=2l+1}. It turns out that these forms can be refined to ε {\displaystyle \varepsilon }-quadratic forms, where ε = ( − 1 ) k {\displaystyle \varepsilon =(-1)^{k}}. These ε {\displaystyle \varepsilon }-quadratic forms define elements in the L-groups L n ( π 1 ( X ) ) {\displaystyle L_{n}(\pi _{1}(X))}.

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

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

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

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

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