Handle disassembly

In differential topology , a branch of mathematics , the handle decomposition is the basis for the classification and description of manifolds .

Definition: gluing on a handle

This 3-dimensional manifold is created by gluing three 1-handles to one 0-handle.

Notation: denote the -dimensional full sphere, the -dimensional sphere. ${\displaystyle B^{n}}$ ${\displaystyle n}$ ${\displaystyle S^{n-1}=\partial B^{n}}$ ${\displaystyle (n-1)}$

In the following, we denote the product as the handle of a -dimensional manifold ${\displaystyle k}$ ${\displaystyle m}$

${\displaystyle H_{k}=B^{k}\times B^{m-k}}$

with the decomposition given by the product structure

${\displaystyle \partial H_{k}=S^{k-1}\times B^{m-k}\cup B^{k}\times S^{m-k-1}}$.

${\displaystyle B^{k}\times \left\{0\right\}}$is called the core and the coke core of the handle. ${\displaystyle \left\{0\right\}\times B^{m-k}}$

Now be a -dimensional differentiable manifold with a boundary . The result of sticking a handle on is manifold ${\displaystyle M}$ ${\displaystyle m}$ ${\displaystyle k}$

${\displaystyle M\cup _{f}H_{k}=\left(M\sqcup (B^{k}\times B^{m-k})\right)/\sim }$

with the equivalence relation generated by for all , ${\displaystyle \sim }$ ${\displaystyle (p,x)\sim f(p,x)}$ ${\displaystyle (p,x)\in S^{k-1}\times B^{m-k}\subset B^{k}\times B^{m-k}}$

for an embedding . A differentiable manifold is obtained by canonically smoothing the corners. (In particular, sticking a -handle is the disjoint union with a -ball ). ${\displaystyle f\colon S^{k-1}\times B^{m-k}\to \partial M}$ ${\displaystyle 0}$ ${\displaystyle m}$ ${\displaystyle B^{m}}$

The manifold obtained in this way is clearly determined by the embedding or, equivalently, by a framed embedding . ${\displaystyle f\colon S^{k-1}\times B^{m-k}\to \partial M}$ ${\displaystyle S^{k-1}\to \partial M}$

The sphere is called the stick-on sphere and the sphere is called the belt sphere . ${\displaystyle S^{k-1}\times \left\{0\right\}}$ ${\displaystyle \left\{0\right\}\times S^{m-k-1}}$

Handle disassembly

Every compact differentiable manifold has a handle decomposition.

The proof of this theorem uses Morse theory . For every differentiated manifold there is a Morse function , the critical points of which correspond to different function values ​​(and do not lie on the edge). The theorem then follows by means of complete induction from the following local description of the vicinity of a critical point. ${\displaystyle M}$ ${\displaystyle f\colon M\to \mathbb {R} }$

Let it be a function with exactly one critical point in and no further critical points in (for a suitable one ). Then arises out by adhering a -Henkels, with the index of the critical point in is. ${\displaystyle f\colon M\to \mathbb {R} }$ ${\displaystyle C^{\infty }}$ ${\displaystyle f^{-1}(0)}$ ${\displaystyle f^{-1}(\left[-\epsilon ,\epsilon \right])}$ ${\displaystyle \epsilon >0}$ ${\displaystyle f^{-1}(\left[-\infty ,\epsilon \right])}$ ${\displaystyle f^{-1}(\left[-\infty ,-\epsilon \right])}$ ${\displaystyle k}$ ${\displaystyle k}$ ${\displaystyle f^{-1}(0)}$

This theorem goes back to Stephen Smale , who sketched a proof in 1961 and then used the Henkel decomposition to prove the Poincaré conjecture in dimensions . John Milnor proved in his book "Morse Theory" a weaker version, which says that it is homotopy equivalent to the space created by gluing a k-cell . Complete proof was given by Palais in 1963. simplified versions can be found at Fukui and Madsen-Tornehave ${\displaystyle \geq 5}$ ${\displaystyle f^{-1}(\left[-\infty ,\epsilon \right])}$ ${\displaystyle f^{-1}(\left[-\infty ,-\epsilon \right])}$

Low dimensional examples

• Classification of the surfaces : Every closed, orientable surface has a handle split up from a 0-handle,1-handleand a 2-handle. The numberis the gender of the surface. ${\displaystyle 2g}$ ${\displaystyle g}$
• Heegaard decomposition of 3-manifolds : A (3-dimensional) handle body of the genderis created by gluing1-handles to a 0-handle. The Heegaard decomposition is the decomposition of a 3-manifold into two handle bodies. Every closed, orientable 3-manifold has a Heegaard decomposition, the minimum possibleis called the Heegaard gender . A Heegaard decomposition determines a handle decomposition of the 3-manifold into a 0-handle,1-handle,2-handle and a 3-handle. ${\displaystyle g}$ ${\displaystyle g}$ ${\displaystyle g}$ ${\displaystyle g}$ ${\displaystyle g}$
• Kirby calculus : Handle decompositions of 4-dimensional manifolds are described by Kirby diagrams .

Relative handle dissection

Let it be a compact, differentiable manifold with a decomposition of the boundary into (possibly empty) subsets ${\displaystyle M}$ ${\displaystyle \partial M}$

${\displaystyle \partial M=\partial _{+}M\sqcup \partial _{-}M}$.

A handle decomposition of relative to is a representation of manifold constructed as by successively gluing handles to . Using Morse theory, one can show that for every such pair there is a handle decomposition of relative to . ${\displaystyle M}$ ${\displaystyle \partial _{-}M}$ ${\displaystyle M}$ ${\displaystyle \partial _{-}M\times \left[0,1\right]}$ ${\displaystyle (M,\partial _{-}M)}$ ${\displaystyle M}$ ${\displaystyle \partial _{-}M}$

Cerf theory

Two handle dismantling of the same manifold can be transformed into one another by handle slide and adding or omitting two complementary handles ( cancellation ).

Handle slide

The manifold arises from gluing on a handle by means of the gluing image . Let it be an isotope with and . Then the manifold constructed by gluing a handle to by means of the gluing image is diffeomorphic to . ${\displaystyle M^{\prime }}$ ${\displaystyle M}$ ${\displaystyle k}$ ${\displaystyle \phi \colon S^{k-1}\times B^{m-k}\to \partial M}$ ${\displaystyle h\colon M\times \left[0,1\right]\to M}$ ${\displaystyle h_{0}=id}$ ${\displaystyle h:=h_{1}}$ ${\displaystyle k}$ ${\displaystyle M}$ ${\displaystyle h\circ \phi }$ ${\displaystyle M^{\prime \prime }}$ ${\displaystyle M^{\prime }}$

In particular, a handle can always be glued in such a way that its adhesive sphere is disjoint from the belt spheres of all handles . As a consequence, one can construct a handle decomposition for every compact, differentiable manifold in such a way that handles are attached to a set of handles in the ascending order of their indices , i.e. H. for which are -Henkel by the glued -Henkeln. ${\displaystyle k}$ ${\displaystyle l}$ ${\displaystyle l\geq k}$ ${\displaystyle 0}$ ${\displaystyle l\geq k}$ ${\displaystyle l}$ ${\displaystyle k}$

Complementary handles

A handle and a handle are called complementary if the adhesive sphere of the handle intersects the belt sphere of the handle transversely at exactly one point. ${\displaystyle k}$ ${\displaystyle (k+1)}$ ${\displaystyle (k+1)}$ ${\displaystyle k}$

If a manifold is created from a manifold by gluing on a -handle and then gluing on a complementary -handle, then it is diffeomorphic to . As a consequence, one can always choose a handle decomposition in such a way that there is exactly one 0-handle and furthermore, if or so that there is exactly one or no handle . ${\displaystyle M^{\prime }}$ ${\displaystyle M}$ ${\displaystyle k}$ ${\displaystyle (k+1)}$ ${\displaystyle M^{\prime }}$ ${\displaystyle M}$ ${\displaystyle \partial M=\emptyset }$ ${\displaystyle \partial M\not =\emptyset }$ ${\displaystyle m}$ ${\displaystyle m=dim(M)}$

Cerf's theorem

Two (relative) handle dissections of a pair (with handles glued on in ascending order of the indices) can be converted into one another by a sequence of handle slides, adding / removing a complementary pair of handles and isotopes. ${\displaystyle (M,\partial _{-}M)}$

Surgery (spherical modifications) and connection to the cobordism theory

2-surgery of the 2-sphere

If a manifold is created from by gluing on a handle, then the (m-1) manifold is created from by surgery , i.e. H. by cutting out the embedded and then pasting in by means of the canonical identification ${\displaystyle M^{\prime }}$ ${\displaystyle M}$ ${\displaystyle k}$ ${\displaystyle \partial M^{\prime }}$ ${\displaystyle \partial M}$ ${\displaystyle (k-1)}$ ${\displaystyle S^{k-1}\times D^{m-k}}$ ${\displaystyle D^{k}\times S^{p-k-1}}$

${\displaystyle \partial (D^{k}\times S^{m-k-1})=S^{k-1}\times S^{m-k-1}=\partial (S^{k-1}\times D^{m-k})}$.

(These surgeries are also referred to as spherical modifications in the literature.)

Let be a cobordism between closed manifolds and , thus a compact manifold with . Then, with Smale's theorem, one obtains a handle decomposition of relative to, and therefore a construction of , through a sequence of operations (spherical modifications). ${\displaystyle W}$ ${\displaystyle M_{0}}$ ${\displaystyle M_{1}}$ ${\displaystyle W}$ ${\displaystyle \partial W=M_{0}\sqcup M_{1}}$ ${\displaystyle W}$ ${\displaystyle M_{0}}$ ${\displaystyle M_{1}}$ ${\displaystyle M_{0}}$

literature

• Robert E. Gompf, András I. Stipsicz: 4-manifolds and Kirby calculus. (= Graduate Studies in Mathematics. 20). American Mathematical Society, Providence, RI 1999, ISBN 0-8218-0994-6 .
• Yukio Matsumoto: An introduction to Morse theory. Translated from the 1997 Japanese original by Kiki Hudson and Masahico Saito. (= Translations of Mathematical Monographs. 208. Iwanami Series in Modern Mathematics). American Mathematical Society, Providence, RI 2002, ISBN 0-8218-1022-7 .

Web links

• Handlebody Decomposition of a Manifold

Individual evidence

1. ↑ Stephen Smale: On the structure of 5-manifolds. In: Ann. of Math. Vol. 75, No. 2, 1962, pp. 38-46.
2. ↑ Stephen Smale: Generalized Poincaré's conjecture in dimensions greater than four. In: Ann. of Math. Vol. 74, No. 2, 1961, pp. 391-406.
3. ^ J. Milnor: Morse theory. Based on lecture notes by M. Spivak and R. Wells. (= Annals of Mathematics Studies. No. 51). Princeton University Press, Princeton, NJ 1963.
4. ^ Richard S. Palais: Morse theory on Hilbert manifolds. In: Topology. 2, 1963, pp. 299-340.
5. ↑ K. Fukui In: Math. Sem. Notes Kobe Univ. Volume 3, no. 1, paper no. X, 1975, pp. 1-4.
6. ↑ Ib Madsen, Jørgen Tornehave: From calculus to cohomology. de Rham cohomology and characteristic classes. Cambridge University Press, Cambridge 1997, ISBN 0-521-58059-5 (Appendix C)
7. ^ J. Milnor: Lectures on the h-Cobordism Theorem. notes by L. Siebenmann and J. Sondow. Princeton University Press, 1974.
8. ↑ Jean Cerf: La stratification naturelle des espaces de fonctions différentiables réelles et le théorème de la pseudo-isotopie. In: Inst. Hautes Études Sci. Publ. Math. No. 39, 1970, pp. 5-173.