Handle theory

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handle-body theory

One of the methods to study topological manifolds, based on the representation of a manifold as a union of topological balls with non-intersecting interiors and with boundaries intersecting in a special way.

Let be an -dimensional manifold, an integer such that , and let a topological ball be the image of a homeomorphic mapping , where denotes the standard -dimensional ball with centre at the point . Then the pair (or, simply, ) is called a handle of index in . The homeomorphism is called the characteristic mapping of the handle , the disc is called the core disc, and the disc is called the transversal disc. The sphere is called the attaching (or base) sphere, and the sphere is called the transversal sphere. The space is called the base of the handle .

If the manifold is piecewise linear, then it makes sense to speak about piecewise-linear handles, meaning piecewise linearity of the characteristic mapping. Similarly, one can speak about smooth handles in the case when is smooth.

Let be a handle of index in the manifold , let be its base and let be a submanifold of such that . The transition from to is called the operation of attaching a handle of index . The imbedding is called the attaching mapping. Up to a homeomorphism invariant of , the manifold is defined entirely by the attaching mapping and is independent of the ambient manifold . If is an arbitrary imbedding of in , then the result of attaching a handle by means of can be described as follows: , where the equivalence relation is generated by identification of the points of and by . The transition from to is called a spherical rearrangement (also called surgery). Manifolds obtained by attaching a handle by means of isotopic imbeddings are homeomorphic. In Fig. athe attaching of three-dimensional handles of indices 1 and 2 is shown. Attaching a handle of index 0 to consists of adding to a separately taken ball of dimension . Adding a handle of index consists of pasting an -dimensional ball to one of the components of .

Figure: h046300a

If the manifold and the attaching imbedding are piecewise linear, then the manifold obtained by attaching a handle by means of is also piecewise linear. In the case of smooth and the manifold has a natural smooth structure at all points except at "corner" points, the union of which coincides with the boundary of the handle's base. This structure can be uniquely extended to a smooth structure on the entire . Such extension is called smoothing of corners. In the smooth case, smoothing of corners is included in attaching a handle. Attaching a smooth handle is completely defined by the attaching sphere with a trivialization of its normal bundle.

The representation of a compact manifold as the union of a finite ordered family of handles in is called a handle decomposition of if each subsequent handle intersects the union of the preceding ones exactly along its base. In other words, admits a handle decomposition if it can be obtained from a ball (or empty set) by sequentially attaching handles. Similarly, by a decomposition of a pair , where is a submanifold of , one understands a representation of as a result of sequentially attaching handles to . In particular, a decomposition of a pair , where is a collar of , is called a handle decomposition of the bordism . A handle decomposition of a non-compact manifold consists of an infinite number of handles. Here it is usually required that the decomposition be locally finite, i.e. each compact set in intersects only a finite number of handles.

By transforming the transversal spheres of the already attached handles and the base sphere of the handle being attached into general position and by replacing the attaching mappings by isotopic ones (cf. Isotopy (in topology)), one may achieve that handles of the same index do not intersect and that the indices of sequentially attached handles do not decrease. Such a handle decomposition is called regular.

Each piecewise-linear manifold can be decomposed into piecewise-linear handles. If is a triangulation of and is its second barycentric subdivision, then as handles of index one can take the closed stars in of the barycentres of -dimensional simplices of (see Fig. b; the definition of a star is given in Complex).

Figure: h046300b

There exists a close relation between smooth handle decompositions of a smooth manifold and smooth functions on with non-degenerate critical points — Morse functions (cf. Morse function). This relation is as follows. Let be a critical point of index of a Morse function such that for some the inverse image of the interval contains no other critical points, and let . Then the manifold is obtained from the manifold by means of attaching a smooth handle of index . Thus, each Morse function on a compact manifold generates a smooth handle decomposition of ; moreover, the number of handles of index in this decomposition coincides with the number of critical points of index . This proves the existence of a smooth handle decomposition for any smooth manifold. Conversely, each smooth handle decomposition of is generated by some Morse function on .

The problem of decomposition of topological manifolds into handles is more complicated. It is known that any closed topological manifold of dimension can be decomposed into topological handles. Manifolds of dimension are combinatorially triangulable and, thus, can be decomposed into handles. It has been proved that there exists a manifold of dimension 4 which does not admit a handle decomposition.

If in the regular handle decomposition of a manifold one contracts sequentially all handles to their core disc, then one obtains a cellular space . To each handle of index in the decomposition of then corresponds a -dimensional cell in the CW-complex of . The space is of the same homotopy type as . If is closed, coincides with .

From the definition of a handle it follows that each -dimensional handle of index is at the same time a handle of index . If is the base of as a handle of index and is the base of as a handle of index , then and . Each handle decomposition of a closed manifold generates a so-called dual handle decomposition of . The dual decomposition consists of the handles of the initial decomposition taken in reverse order, moreover, each handle of index is considered already as a handle of index . This fact is the geometric foundation for Poincaré duality. If the manifold has a boundary, then the dual decomposition can be considered as a decomposition of the pair .

Let be non-intersecting handles of index attached to a manifold with a simply-connected boundary by imbeddings . Let and and let denote the element of the group determined by . Then the imbedding is isotopic in to an imbedding such that . This means that the manifolds and , where is the handle attached by means of , are homeomorphic. The transition from the manifold to the manifold is called addition of handles.

Let a manifold be obtained from a manifold by subsequently attaching a handle of index and a handle of index so that the base sphere of cuts transversally the transversal sphere of in exactly one point. Then this pair of handles can be removed. This means that there exists a homeomorphism of onto which is the identity outside a neighbourhood of . This operation of removing is sometimes called cancellation of additional handles. Addition of handles and cancellation of handles can be carried out while staying within the piecewise-linear or the smooth category. By cancellation of the handles of indices 0 and 1 one can, for example, replace any handle decomposition of a compact connected manifold by the handle decomposition with exactly one handle of index 0. If is simply connected and , then by addition and cancellation of handles one can reduce any handle decomposition to a decomposition with a minimal number of handles compatible with the homological structure of .

Let be a topological handle of index in a piecewise-linear manifold ; moreover, let the characteristic mapping be piecewise linear in a neighbourhood of . Is there an isotopy of that is the identity in a neighbourhood of and which straightens the handle , i.e. transforms it into a piecewise-linear handle? If the answer to this question is always positive, then on each topological manifold one can introduce a piecewise-linear structure by making the structures on the coordinate neighbourhood compatible using straightening piecewise-linear handles of one neighbourhood inside another. In reality the answer depends on the index and the dimension of the handle . If or and , then any handle can be straightened. It is known that when there exists handles of index 3 that cannot be straightened; moreover, the obstruction to straightening lies in the group . In dimension 4, the handles of indices 0 and 1 can be straightened, and those of indices 2 and 3, in general, can not. It are the so-called Milnor groups that form obstructions to smoothing piecewise-linear handles.

References

 [1] S. Smale, "Generalised Poincaré conjecture in dimensions " Ann. of Math (2) , 74 (1961) pp. 391–466 [2] S. Smale, "Structure of manifolds" Amer. J. Math. , 84 (1962) pp. 387–399 [3] S. Smale, "A survey of some recent developments in differential topology" Bull. Amer. Math. Soc. , 69 (1963) pp. 131–145 [4] R.C. Kirby, L.C. Siebenmann, "Foundational essays on topological manifolds, smoothings, and triangulations" , Princeton Univ. Press (1977) [5] C.P. Rourke, B.J. Sanderson, "Introduction to piecewise-linear topology" , Springer (1972) [6] V.A. Rokhlin, D.B. Fuks, "Beginner's course in topology. Geometric chapters" , Springer (1984) (Translated from Russian)