Heegaard decomposition

In mathematics , Heegaard decompositions are an important aid for 3-dimensional topology . They are named after the Danish mathematician Poul Heegaard .

definition

A Heegaard decomposition of a closed 3-dimensional manifold consists of two handle bodies and and a homeomorphism , so that from and by gluing means arises, i.e. H. one has a homeomorphism ${\displaystyle M}$ ${\displaystyle H_{1}}$ ${\displaystyle H_{2}}$ ${\displaystyle f:\partial H_{1}\rightarrow \partial H_{2}}$ ${\displaystyle M}$ ${\displaystyle H_{1}}$ ${\displaystyle H_{2}}$ ${\displaystyle f}$

${\displaystyle M\cong (H_{1}\cup H_{2})/{\sim }}$

for the through

${\displaystyle x\sim y\Longleftrightarrow y=f(x),x\in \partial H_{1},y\in \partial H_{2}}$

given relation.

The gender of the surfaces is called the gender of the Heegaard decomposition. The area embedded in is called the Heegaard area of the Heegaard decomposition. ${\displaystyle \partial H_{1}\cong \partial H_{2}}$ ${\displaystyle M}$ ${\displaystyle \partial H_{1}=\partial H_{2}\subset M}$

The Heegaard gender is the minimum gender across all Heegaard decompositions of . The Heegaard Euler characteristic is the negative of the maximum of the Euler characteristic of all Heegaard surfaces, ie . ${\displaystyle g(M)}$ ${\displaystyle M}$ ${\displaystyle \chi _{-}^{h}(M)}$ ${\displaystyle \chi _{-}^{h}(M)=2g(M)-2}$

The Heegaard gradient of is the infimum over all finite overlaps of , where denotes the degree of the overlap . ${\displaystyle M}$ ${\displaystyle \inf {\frac {\chi _{-}^{h}(M_{i})}{d_{i}}}}$ ${\displaystyle M}$ ${\displaystyle d_{i}}$ ${\displaystyle M_{i}\to M}$

existence

From the Morse theory it follows that every closed orientable 3-manifold has a Heegaard decomposition. Alternatively, the existence of Heegaard decompositions also results from the triangulability of 3-manifolds, one can choose the environment of the 1-skeleton of a triangulation as the handle body, its complement is then also a handle body as the environment of the 1-skeleton of the dual triangulation.

Examples

• Standard Heegaard decomposition of the 3-sphere : Let handle body be of gender (i.e. full spheres) and then is . ${\displaystyle H_{1},H_{2}}$ ${\displaystyle 0}$ ${\displaystyle f=id:S^{2}\rightarrow S^{2}}$ ${\displaystyle (H_{1}\cup H_{2})/{\sim }\cong S^{3}}$
• Let handle body be of gender (i.e. full tori) and then is . ${\displaystyle H_{1},H_{2}}$ ${\displaystyle 1}$ ${\displaystyle f=id:T^{2}\rightarrow T^{2}}$ ${\displaystyle (H_{1}\cup H_{2})/{\sim }\cong S^{2}\times S^{1}}$
• Gender-1-Heegaard decomposition of the 3-sphere : Be the handle body of the gender and map the longitude to the meridian and the meridian to the longitude, then is . ${\displaystyle H_{1},H_{2}}$ ${\displaystyle 1}$ ${\displaystyle f:T^{2}\rightarrow T^{2}}$ ${\displaystyle (H_{1}\cup H_{2})/{\sim }\cong S^{3}}$
• Standard Heegaard decomposition of the lens spaces : Let the handle body be of the gender and be given by an arbitrary matrix , then there is a lens space . ${\displaystyle H_{1},H_{2}}$ ${\displaystyle 1}$ ${\displaystyle f:T^{2}\rightarrow T^{2}}$ ${\displaystyle A\in SL(2,\mathbb {Z} )}$ ${\displaystyle (H_{1}\cup H_{2})/{\sim }\cong L(p,q)}$
• Heegaard decomposition of surface bundles : Every surface bundle with a fiber of the gender has a Heegaard decomposition of the gender . In particular, the Heegaard gradient is a bundle of areas . Since, according to Agol's theorem, every 3-manifold is finitely overlaid by a bundle of surfaces, the Heegaard gradient is always trivial. ${\displaystyle g}$ ${\displaystyle 2g+1}$ ${\displaystyle 0}$

Stabilizations, reducibility, irreducibility

From a Heegaard decomposition of a manifold one can obtain further Heegard decompositions of the same 3-manifold with Heegaard surfaces of the higher sex through stabilization (gluing on additional handles, for which longitudes are mapped to meridians and meridians to longitudes). These Heegaard decompositions obtained by stabilization are reducible ; H. There is a closed curve in the Heegaard surface, which borders a circular disk in both handle bodies (but not in the Heegaard surface). A Heegaard decomposition is called irreducible if there is no such curve. The lemma hook states that Heegaard decompositions a reducible 3-manifold are always reducible.

A Heegaard decomposition is called weakly reducible if there are two disjoint (not null-homotopic) closed curves in the Heegaard surface, which border circular disks in different handle bodies of the Heegaard decomposition. Otherwise the Heegaard decomposition is called strongly irreducible . Casson and Gordon proved in 1987 that all irreducible Heegaard decompositions are strongly irreducible.

Manifolds with a margin

For a 3- manifold with boundary is defined Heegaard decompositions analogously as decompositions in two compression body with . ${\displaystyle M}$ ${\displaystyle M=H_{1}\cup H_{2}}$ ${\displaystyle \partial _{+}H_{1}\cup \partial _{+}H_{2}}$

A generalized Heegaard decomposition of is a decomposition into (not necessarily connected ) compression bodies and surfaces with and . The union of the compression bodies must be whole and their inner cores should be disjoint. ${\displaystyle M}$ ${\displaystyle V_{i},W_{i},i=1,\dots ,n}$ ${\displaystyle H_{i},i=1,\dots ,n}$ ${\displaystyle \partial _{+}V_{i}=\partial _{+}W_{i}=H_{i}}$ ${\displaystyle \partial _{-}W_{i}=\partial _{-}V_{i+1}}$ ${\displaystyle M}$

literature

• Saveliev, Nikolai: Lectures on the topology of 3-manifolds. An introduction to the Casson invariant. Second revised edition. de Gruyter Textbook. Walter de Gruyter & Co., Berlin, 2012. ISBN 978-3-11-025035-0

Web links

• Jesse Johnson: Notes on Heegaard splittings

Individual evidence

1. ↑ P. Heegaard: Forstudier til en topologisk teori for de algebraiske fladers sammenhaeng , dissertation, Copenhagen 1898.