Surgery obstruction

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

In mathematics, specifically in surgery theory, the surgery obstructions define a map [math]\displaystyle{ \theta \colon \mathcal{N} (X) \to L_n (\pi_1 (X)) }[/math] 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 [math]\displaystyle{ n \geq 5 }[/math]:

A degree-one normal map [math]\displaystyle{ (f,b) \colon M \to X }[/math] is normally cobordant to a homotopy equivalence if and only if the image [math]\displaystyle{ \theta (f,b)=0 }[/math] in [math]\displaystyle{ L_n (\mathbb{Z} [\pi_1 (X)]) }[/math].

Sketch of the definition

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

Consider a degree-one normal map [math]\displaystyle{ (f,b) \colon M \to X }[/math]. The idea in deciding the question whether it is normally cobordant to a homotopy equivalence is to try to systematically improve [math]\displaystyle{ (f,b) }[/math] so that the map [math]\displaystyle{ f }[/math] becomes [math]\displaystyle{ m }[/math]-connected (that means the homotopy groups [math]\displaystyle{ \pi_* (f)=0 }[/math] for [math]\displaystyle{ * \leq m }[/math]) for high [math]\displaystyle{ m }[/math]. It is a consequence of Poincaré duality that if we can achieve this for [math]\displaystyle{ m \gt \lfloor n/2 \rfloor }[/math] then the map [math]\displaystyle{ f }[/math] already is a homotopy equivalence. The word systematically above refers to the fact that one tries to do surgeries on [math]\displaystyle{ M }[/math] to kill elements of [math]\displaystyle{ \pi_i (f) }[/math]. In fact it is more convenient to use homology of the universal covers to observe how connected the map [math]\displaystyle{ f }[/math] is. More precisely, one works with the surgery kernels [math]\displaystyle{ K_i (\tilde M) : = \mathrm{ker} \{f_* \colon H_i (\tilde M) \rightarrow H_i (\tilde X)\} }[/math] which one views as [math]\displaystyle{ \mathbb{Z}[\pi_1 (X)] }[/math]-modules. If all these vanish, then the map [math]\displaystyle{ f }[/math] is a homotopy equivalence. As a consequence of Poincaré duality on [math]\displaystyle{ M }[/math] and [math]\displaystyle{ X }[/math] there is a [math]\displaystyle{ \mathbb{Z}[\pi_1 (X)] }[/math]-modules Poincaré duality [math]\displaystyle{ K^{n-i} (\tilde M) \cong K_i (\tilde M) }[/math], so one only has to watch half of them, that means those for which [math]\displaystyle{ i \leq \lfloor n/2 \rfloor }[/math].

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

1. If [math]\displaystyle{ n=2k }[/math] then the only nontrivial homology group is the kernel [math]\displaystyle{ K_k (\tilde M) : = \mathrm{ker} \{f_* \colon H_k (\tilde M) \rightarrow H_k (\tilde X)\} }[/math]. It turns out that the cup-product pairings on [math]\displaystyle{ M }[/math] and [math]\displaystyle{ X }[/math] induce a cup-product pairing on [math]\displaystyle{ K_k(\tilde M) }[/math]. This defines a symmetric bilinear form in case [math]\displaystyle{ k=2l }[/math] and a skew-symmetric bilinear form in case [math]\displaystyle{ k=2l+1 }[/math]. It turns out that these forms can be refined to [math]\displaystyle{ \varepsilon }[/math]-quadratic forms, where [math]\displaystyle{ \varepsilon = (-1)^k }[/math]. These [math]\displaystyle{ \varepsilon }[/math]-quadratic forms define elements in the L-groups [math]\displaystyle{ L_n (\pi_1 (X)) }[/math].

2. If [math]\displaystyle{ n=2k+1 }[/math] 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 [math]\displaystyle{ L_n (\pi_1 (X)) }[/math].

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

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

If [math]\displaystyle{ n = 4l+2 }[/math] then the surgery obstruction is the Arf-invariant of the associated kernel quadratic form over [math]\displaystyle{ \mathbb{Z}_2 }[/math].

References



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