Lens room

Lens spaces are geometrical structures that occur in mathematics primarily in 3-dimensional topology . They are the simplest class of 3-dimensional closed manifolds . Heinrich Tietze described her for the first time in 1908 . With the lens spaces introduced by Tietze, James Waddell Alexander succeeded in 1919 in refuting a conjecture by Henri Poincaré , since they provide examples of non-homeomorphic spaces with the same fundamental group. Furthermore, lens spaces were the first examples of homotopy-equivalent , but not homeomorphic manifolds: Kurt Reidemeister developed the Reidemeister torsion , later named after him, in 1935 in order to differentiate the homeomorphism type from lens spaces.

definition

Be for natural numbers so that for all . The lens space is defined as the orbit space given by the formula ${\displaystyle m,l_{j}}$ ${\displaystyle j=1,\ldots ,d}$ ${\displaystyle ggT(l_{j},m)=1}$ ${\displaystyle j}$ ${\displaystyle L(m;l_{1},\ldots ,l_{d})}$

${\displaystyle \mathbb {Z} /m\mathbb {Z} \times S^{2d-1}\to S^{2d-1}}$

${\displaystyle (k,(z_{1},\ldots ,z_{d}))\mapsto (z_{1}\cdot e^{2\pi ikl_{1}/m},\ldots ,z_{d}\cdot e^{2\pi ikl_{d}/m})}$

given free action of the cyclic group on the unit sphere . ${\displaystyle \mathbb {Z} /m\mathbb {Z} }$ ${\displaystyle S^{2d-1}\subset \mathbb {C} ^{d}}$

Invariants

The fundamental group of the lens space is independent of . ${\displaystyle L(m;l_{1},\ldots ,l_{d})}$ ${\displaystyle \mathbb {Z} /m\mathbb {Z} }$ ${\displaystyle l_{1},\ldots ,l_{d}}$

The homology groups are calculated as follows:

${\displaystyle H_{0}(L)=\mathbb {Z} ,H_{2d-1}(L)=\mathbb {Z} ,H_{2i-1}(L)=\mathbb {Z} /m\mathbb {Z} }$for , for everyone else . ${\displaystyle 1\leq i\leq d-1}$ ${\displaystyle H_{i}(L)=0}$ ${\displaystyle i}$

classification

Because the fundamental group is lens space , two lens spaces can only be homotopy equivalent if the number matches. ${\displaystyle \mathbb {Z} /m\mathbb {Z} }$ ${\displaystyle m}$

The lens chambers and are ${\displaystyle L(m;l_{1},\ldots ,l_{d})}$ ${\displaystyle L(m;l_{1}^{\prime },\ldots ,l_{d}^{\prime })}$

• homotopy equivalent if and only for a .
  ${\displaystyle l_{1}\ldots l_{d}\equiv \pm n^{d}l_{1}^{\prime }\ldots l_{d}^{\prime }{\pmod {m}}}$
  ${\displaystyle n\in \mathbb {N} }$
• homeomorphic if and only if there is a permutation and a such that for . ${\displaystyle \sigma }$ ${\displaystyle n\in \mathbb {N} }$
  ${\displaystyle l_{j}\equiv \pm nl_{\sigma (j)}{\pmod {m}}}$ ${\displaystyle j=1,\ldots ,d}$

3-dimensional lens spaces

3-dimensional lens spaces are the only 3-manifolds that have a Heegaard decomposition of the gender . ${\displaystyle 1}$

They are spherical 3-manifolds: their universal superposition is the 3-sphere . In particular, they have a Riemannian metric of constant positive sectional curvature .

literature

• Ralph Stöcker, Heiner Zieschang : Algebraic Topology. An introduction. Second edition. Math guides. BG Teubner, Stuttgart 1994, ISBN 3-519-12226-X
• Nikolai Saveliev: 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
• Claude Weber: Lens spaces among 3-manifolds and quotient surface singularities , RACSAM 112, 2018, pp. 893-914.

Web links

• Lens Spaces (Manifold Atlas)

Individual evidence

1. ^ Jean Dieudonné: A history of algebraic and differential topology, 1900-1960 (=  Modern Birkhauser Classics ). Reprint of the 1989 edition. Birkhäuser , Boston 2009, ISBN 978-0-8176-4906-7 , p. 42 , doi : 10.1007 / 978-0-8176-4907-4 .
2. ↑ H. Tietze: About the topological invariants of multidimensional manifolds . In: Monthly books for mathematics and physics . tape 19 , no. 1 , December 1908, ISSN  0026-9255 , p. 1-118 , doi : 10.1007 / BF01736688 .
3. ^ Kurt Reidemeister: Homotopy rings and lens rooms . Dep. Math. Sem. Univ. Hamburg 11: 102-109 (1935)
4. ↑ P. Olum: Mappings of manifolds and the notion of degree . In: Ann. of Math. , (2) 58, 1953, pp. 458-480. JSTOR 1969748
5. ^ EJ Brody: The topological classification of the lens spaces . In: Ann. of Math. , (2) 71, 1960, pp. 163-184. JSTOR 1969884
6. ^ John Milnor : Whitehead torsion . In: Bull. Amer. Math. Soc. , 72, 1966, pp. 358-426. maths.ed.ac.uk ( Memento of the original from May 29, 2016 in the Internet Archive ; PDF) Info: The archive link was inserted automatically and has not yet been checked. Please check the original and archive link according to the instructions and then remove this notice.