page_title: Surgery theory
section_title: Examples
hierarchical_section_title: Surgery theory / Surgery on a manifold / Examples
caption_attribution_description: English: Surgery applied to a two-sphere, removing a cylinder and inserting two disks.
context_page_description: In mathematics, specifically in geometric topology, surgery theory is a collection of techniques used to produce one finite-dimensional manifold from another in a 'controlled' way, introduced by John Milnor. Originally developed for differentiable manifolds, surgery techniques also apply to piecewise linear and topological 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. It is a major tool in the study and classification of manifolds of dimension greater than 3.
  More technically, the idea is to start with a well-understood manifold M and perform surgery on it to produce a manifold M ′ having some desired property, in such a way that the effects on the homology, homotopy groups, or other invariants of the manifold are known.
  The classification of exotic spheres by Michel Kervaire and Milnor led to the emergence of surgery theory as a major tool in high-dimensional topology.
context_section_description: Surgery on the circle
  As per the above definition, a surgery on the circle consists of cutting out a copy of S⁰ × D¹ and gluing in D¹ × S⁰. The pictures in Fig. 1  show that the result of doing this is either (i) S¹ again, or (ii) two copies of S¹.
  Surgery on the 2-sphere
  In this case there are more possibilities, since we can start by cutting out either S¹ × D¹ or S⁰ × D².
  S¹ × D¹: If we remove a cylinder from the 2-sphere, we are left with two disks. We have to glue back in S⁰ × D² – that is, two disks – and it is clear that the result of doing so is to give us two disjoint spheres. (Fig. 2a)
  S⁰ × D²: Having cut out two disks S⁰ × D², we glue back in the cylinder S¹ × D¹. 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 the torus S¹ × S¹, but if they are different, we obtain the Klein Bottle (Fig. 2c).
  Surgery on the n-sphere
  If n = p + q, then . The p-surgery on Sⁿ is therefore . Examples 1 and 2 above were a special case of this.
  Morse functions
  
  Suppose that f is a Morse function on an (n + 1)-dimensional manifold, and suppose that c 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  is obtained from  by a p-surgery. The bordism  can be identified with the trace of this surgery.
  
  Indeed, in some coordinate chart around the critical point, the function f is of the form , with , and p + q + 1 = n + 1. Fig. 3 shows, in this local chart, the manifold M in blue and the manifold M′ in red. The colored region between M and M′ corresponds to the bordism W. The picture shows that W is diffeomorphic to the union
  
  
  (neglecting the issue of straightening corners), where M × I is colored in yellow, and  is colored in green. The manifold M′, being a boundary component of W, is therefore obtained from M by a p-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 manifold M may be regarded as a bordism from the boundary ∂M (which may be empty) to the empty manifold, and so may be obtained from ∂M × I by attaching handles.
caption: Fig. 2a
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