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Surgery theory

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Techniques in topology used to produce one finite-dimensional manifold from another

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Inmathematics, specifically ingeometric topology,surgery theory is a collection of techniques used to produce one finite-dimensionalmanifold from another in a 'controlled' way, introduced byJohn Milnor (1961). Milnor called this techniquesurgery, whileAndrew Wallace called itspherical modification.^[1] The "surgery" on adifferentiable manifoldM of dimension $n=p+q+1$ , could be described as removing an imbeddedsphere of dimensionp fromM.^[2] Originally developed for differentiable (or,smooth) manifolds, surgery techniques also apply topiecewise linear (PL-) andtopological manifolds.

Surgery refers to cutting out parts of the manifold and replacing it with a part of another manifold, matching up along the cut or boundary. This is closely related to, but not identical with,handlebody decompositions.

More technically, the idea is to start with a well-understood manifoldM and perform surgery on it to produce a manifoldM′ having some desired property, in such a way that the effects on thehomology,homotopy groups, or other invariants of the manifold are known. A relatively easy argument usingMorse theory shows that a manifold can be obtained from another one by a sequence of spherical modificationsif and only if those two belong to the samecobordism class.^[1]

The classification ofexotic spheres byMichel Kervaire and Milnor (1963) led to the emergence of surgery theory as a major tool in high-dimensional topology.

Surgery on a manifold

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A basic observation

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IfX,Y are manifolds with boundary, then the boundary of the product manifold is

\partial (X\times Y)=(\partial X\times Y)\cup (X\times \partial Y).

The basic observation which justifies surgery is that the space $S^{p}\times S^{q-1}$ can be understood either as the boundary of $D^{p+1}\times S^{q-1}$ or as the boundary of $S^{p}\times D^{q}$ . In symbols,

\partial \left(S^{p}\times D^{q}\right)=S^{p}\times S^{q-1}=\partial \left(D^{p+1}\times S^{q-1}\right)

where $D^{q}$ is theq-dimensional disk, i.e., the set of points in $\mathbb {R} ^{q}$ that are at distance one-or-less from a given fixed point (the center of the disk); for example, then, $D^{1}$ ishomeomorphic to the unit interval, while $D^{2}$ is a circle together with the points in its interior.

Surgery

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Now, given a manifoldM of dimension $n=p+q$ and anembedding $\phi \colon S^{p}\times D^{q}\to M$ , define anothern-dimensional manifold $M^{'} {\displaystyle M'}$ to be

M':=\left(M\setminus \operatorname {int} (\operatorname {im} (\phi ))\right)\;\cup _{\phi |_{S^{p}\times S^{q-1}}}\left(D^{p+1}\times S^{q-1}\right).

Since $\operatorname {im} (\phi )=\phi (S^{p}\times D^{q})$ and from the equation from our basic observation before, the gluing is justified then

\phi \left(\partial \left(S^{p}\times D^{q}\right)\right)=\phi \left(S^{p}\times S^{q-1}\right).

One says that the manifoldM′ is produced by asurgery cutting out $S^{p}\times D^{q}$ and gluing in $D^{p+1}\times S^{q-1}$ , or by ap-surgery if one wants to specify the numberp. Strictly speaking,M′ is a manifold with corners, but there is a canonical way to smooth them out. Notice that thesubmanifold that was replaced inM was of the same dimension asM (it was ofcodimension 0).

Attaching handles and cobordisms

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Surgery is closely related to (but not the same as)handle attaching. Given an $(n+1)$ -manifold with boundary $(L,\partial L)$ and an embedding $\phi \colon S^{p}\times D^{q}\to \partial L$ , where $n=p+q$ , define another $(n+1)$ -manifold with boundaryL′ by

L':=L\;\cup _{\phi }\left(D^{p+1}\times D^{q}\right).

The manifoldL′ is obtained by "attaching a $(p+1)$ -handle", with $\partial L'$ obtained from $\partial L$ by ap-surgery

\partial L'=(\partial L\setminus \operatorname {int} (\operatorname {im} (\phi )))\;\cup _{\phi |_{S^{p}\times D^{q}}}\left(D^{p+1}\times S^{q-1}\right).

A surgery onM not only produces a new manifoldM′, but also acobordismW betweenM andM′. Thetrace of the surgery is thecobordism $(W;M,M')$ , with

W:=(M\times I)\;\cup _{\phi \times \{1\}}\left(D^{p+1}\times D^{q}\right)

the $(n+1)$ -dimensional manifold with boundary $\partial W=M\cup M'$ obtained from the product $M\times I$ by attaching a $(p+1)$ -handle $D^{p+1}\times D^{q}$ .

Surgery is symmetric in the sense that the manifoldM can be re-obtained fromM′ by a $(q-1)$ -surgery, the trace of which coincides with the trace of the original surgery, up to orientation.

In most applications, the manifoldM comes with additional geometric structure, such as a map to some reference space, or additional bundle data. One then wants the surgery process to endowM′ with the same kind of additional structure. For instance, a standard tool in surgery theory is surgery onnormal maps: such a process changes a normal map to another normal map within the same bordism class.

Examples

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Surgery on the circle
Fig. 1
As per the above definition, a surgery on the circle consists of cutting out a copy of $S^{0}\times D^{1}$ and gluing in $D^{1}\times S^{0}$ . The pictures in Fig. 1 show that the result of doing this is either (i) $S^{1}$ again, or (ii) two copies of $S^{1}$ .
Fig. 2a
Fig. 2b
Surgery on the 2-sphere
In this case there are more possibilities, since we can start by cutting out either $S^{1}\times D^{1}$ or $S^{0}\times D^{2}$ .
1. $S^{1}\times D^{1}$ : If we remove a cylinder from the 2-sphere, we are left with two disks. We have to glue back in $S^{0}\times D^{2}$ – that is, two disks – and it is clear that the result of doing so is to give us two disjoint spheres. (Fig. 2a)
  Fig. 2c. This shape cannot be embedded in 3-space.
2. $S^{0}\times D^{2}$ : Having cut out two disks $S^{0}\times D^{2}$ , we glue back in the cylinder $S^{1}\times D^{1}$ . There are two possible outcomes, depending on whether our gluing maps have the same or opposite orientation on the two boundary circles. If the orientations are the same (Fig. 2b), the resulting manifold is thetorus $S^{1}\times S^{1}$ , but if they are different, we obtain theKlein bottle (Fig. 2c).
Surgery on then-sphere
If $n=p+q$ , then
$S^{n}=\partial D^{n+1}\approx \partial (D^{p+1}\times D^{q})=S^{p}\times D^{q}\;\cup \;D^{p+1}\times S^{q-1}$ .
Thep-surgery on ' $S^{n}$ is therefore
$D^{p+1}\times S^{q-1}\;\cup \;D^{p+1}\times S^{q-1}=S^{p+1}\times S^{q-1}$ .
Examples 1 and 2 above were a special case of this.
Morse functionsSuppose thatf is aMorse function on an (n + 1)-dimensional manifold, and suppose thatc is a critical value with exactly one critical point in its preimage. If the index of this critical point is $p+1$ , then the level-set $M':=f^{-1}(c+\varepsilon )$ is obtained from $M:=f^{-1}(c-\varepsilon )$ by ap-surgery. The bordism $W:=f^{-1}([c-\varepsilon ,c+\varepsilon ])$ can be identified with the trace of this surgery.Indeed, in some coordinate chart around the critical point, the functionf is of the form $-\Vert x\Vert ^{2}+\Vert y\Vert ^{2}$ , with $x\in R^{p+1},y\in R^{q}$ , and $p+q+1=n+1$ .Fig. 3 shows, in this local chart, the manifoldM in blue and the manifoldM′ in red. The colored region betweenM andM′ corresponds to the bordismW. The picture shows thatW is diffeomorphic to the union
$W\cong M\times I\cup _{S^{p}\times D^{q}}D^{p+1}\times D^{q}$
(neglecting the issue of straightening corners), where $M\times I$ is colored in yellow, and $D^{p+1}\times D^{q}$ is colored in green. The manifoldM′, being a boundary component ofW, is therefore obtained fromM by ap-surgery. Since every bordism between closed manifolds has a Morse function where different critical points have different critical values, this shows that any bordism can be decomposed into traces of surgeries (handlebody decomposition). In particular, every manifoldM may be regarded as a bordism from the boundary ∂M (which may be empty) to the empty manifold, and so may be obtained from $\partial M\times I$ by attaching handles.

Effects on homotopy groups, and comparison to cell-attachment

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Intuitively, the process of surgery is the manifold analog of attaching a cell to a topological space, where the embedding $\phi$ takes the place of the attaching map. A simple attachment of a $(p+1)$ -cell to ann-manifold would destroy the manifold structure for dimension reasons, so it has to be thickened by crossing with another cell.

Up to homotopy, the process of surgery on an embedding $\phi \colon S^{p}\times D^{q}\to M$ can be described as the attaching of a $(p+1)$ -cell, giving the homotopy type of the trace, and the detaching of aq-cell to obtainN. The necessity of the detaching process can be understood as an effect ofPoincaré duality.

In the same way as a cell can be attached to a space to kill an element in somehomotopy group of the space, ap-surgery on a manifoldM can often be used to kill an element $\alpha \in \pi _{p}(M)$ . Two points are important however: Firstly, the element $\alpha \in \pi _{p}(M)$ has to be representable by an embedding $\phi \colon S^{p}\times D^{q}\to M$ (which means embedding the corresponding sphere with a trivialnormal bundle). For instance, it is not possible to perform surgery on an orientation-reversing loop. Secondly, the effect of the detaching process has to be considered, since it might also have an effect on the homotopy group under consideration. Roughly speaking, this second point is only important whenp is at least of the order of half the dimension of M.

Application to classification of manifolds

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The origin and main application of surgery theory lies in theclassification of manifolds of dimension greater than four. Loosely, the organizing questions of surgery theory are:

IsX a manifold?
Isf a diffeomorphism?

More formally, one asks these questionsup tohomotopy:

Does a spaceX have the homotopy type of a smooth manifold of a given dimension?
Is ahomotopy equivalence $f\colon M\to N$ between two smooth manifoldshomotopic to a diffeomorphism?

It turns out that the second ("uniqueness") question is a relative version of a question of the first ("existence") type; thus both questions can be treated with the same methods.

Note that surgery theory doesnot give acomplete set of invariants to these questions. Instead, it isobstruction-theoretic: there is a primary obstruction, and a secondary obstruction called thesurgery obstruction which is only defined if the primary obstruction vanishes, and which depends on the choice made in verifying that the primary obstruction vanishes.

The surgery approach

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In the classical approach, as developed byWilliam Browder,Sergei Novikov,Dennis Sullivan, andC. T. C. Wall, surgery is done onnormal maps of degree one. Using surgery, the question "Is the normal map $f\colon M\to X$ of degree one cobordant to a homotopy equivalence?" can be translated (in dimensions greater than four) to an algebraic statement about some element in anL-group of thegroup ring $\mathbb {Z} [\pi _{1}(X)]$ . More precisely, the question has a positive answer if and only if thesurgery obstruction $\sigma (f)\in L_{n}(\mathbb {Z} [\pi _{1}(X)])$ is zero, wheren is the dimension ofM.

For example, consider the case where the dimensionn = 4k is a multiple of four, and $\pi _{1}(X)=0$ . It is known that $L_{4k}(\mathbb {Z} )$ is isomorphic to the integers $\mathbb {Z}$ ; under this isomorphism the surgery obstruction off is proportional to the difference of thesignatures $\sigma (X)-\sigma (M)$ ofX andM. Hence a normal map of degree one is cobordant to a homotopy equivalence if and only if the signatures of domain and codomain agree.

Coming back to the "existence" question from above, we see that a spaceX has the homotopy type of a smooth manifold if and only if it receives a normal map of degree one whose surgery obstruction vanishes. This leads to a multi-step obstruction process: In order to speak of normal maps,X must satisfy an appropriate version ofPoincaré duality which turns it into aPoincaré complex. Supposing thatX is a Poincaré complex, thePontryagin–Thom construction shows that a normal map of degree one toX exists if and only if theSpivak normal fibration ofX has a reduction to astable vector bundle. If normal maps of degree one toX exist, their bordism classes (callednormal invariants) are classified by the set of homotopy classes $[X,G/O]$ . Each of these normal invariants has a surgery obstruction;X has the homotopy type of a smooth manifold if and only if one of these obstructions is zero. Stated differently, this means that there is a choice of normal invariant with zero image under thesurgery obstruction map

[X,G/O]\to L_{n}\left(\mathbb {Z} \left[\pi _{1}(X)\right]\right).

Structure sets and surgery exact sequence

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The concept ofstructure set is the unifying framework for both questions of existence and uniqueness. Roughly speaking, the structure set of a spaceX consists of homotopy equivalencesM →X from some manifold toX, where two maps are identified under a bordism-type relation. A necessary (but not in general sufficient) condition for the structure set of a spaceX to be non-empty is thatX be ann-dimensional Poincaré complex, i.e. that thehomology andcohomology groups be related by isomorphisms $H^{*}(X)\cong H_{n-*}(X)$ of ann-dimensional manifold, for some integern. Depending on the precise definition and the category of manifolds (smooth,PL, ortopological), there are various versions of structure sets. Since, by thes-cobordism theorem, certain bordisms between manifolds are isomorphic (in the respective category) to cylinders, the concept of structure set allows a classification even up todiffeomorphism.

The structure set and the surgery obstruction map are brought together in thesurgery exact sequence. This sequence allows to determine the structure set of a Poincaré complex once the surgery obstruction map (and a relative version of it) are understood. In important cases, the smooth or topological structure set can be computed by means of the surgery exact sequence. Examples are the classification ofexotic spheres, and the proofs of theBorel conjecture fornegatively curved manifolds and manifolds withhyperbolic fundamental group.

In the topological category, the surgery exact sequence is the long exact sequence induced by afibration sequence ofspectra. This implies that all the sets involved in the sequence are in fact abelian groups. On the spectrum level, the surgery obstruction map is anassembly map whose fiber is the block structure space of the corresponding manifold.