manifolds and cobordisms
cobordism theory, Introduction
Definitions
Genera and invariants
Classification
Theorems
A lens space is the quotient space
of a (2d+1)-dimensional sphere by a free action of a finite cyclic group $\mathbb{Z}_n$; specifically — under the identification of $S^{2d+1}$ with the unit sphere in the real vector space underlying $\mathbb{C}^{d+1}$ — by the action induced from a linear representation of the form
for natural numbers $l_i$ coprime to $n$: $gcd(l_i,n) = 1$.
Often this is considered by default for $2d + 1 = 3$, in which case lens spaces are examples of 3-manifolds, and as such are in some sense the simplest after the 3-sphere $S^{3}$ and the product manifold $S^{2} \times S^{1}$ of the 2-sphere with the circle.
Two lens spaces $L\big(n,\{l_i\}_{i=0}^d\big)$ and $L\big(n,\{l'_i\}_{i=0}^d\big)$ are homotopy equivalent iff there exists a $k\in\mathbb{N}$ with
Two lens spaces $L\big(n,\{l_i\}_{i=0}^d\big)$ and $L\big(n,\{l'_i\}_{i=0}^d\big)$ are homeomorphic iff there exists a $k\in\mathbb{N}$ and a permutation $\sigma\in Sym_d$ with
for all $i=0,\ldots, d$.
In both theorems, the same $n$ for both lens spaces is a necessary condition for them to be homeomorphic or homotopy equivalent as follows with their respective fundamental group.
Lens spaces are the only 3-manifolds with finite, cyclic, non-trivial fundamental group. The reverse direction of this statement (that a 3-manifold with such a fundamental group must be a lens space) is closely related to the Poincaré conjecture.
Lens spaces are important in geometric topology. They offer the simplest examples of smooth manifolds which cannot be distinguished by homotopy theory.
The 3-dimensional lens space $L(p,q) \coloneqq L(p, \{1,q\})$, for coprime integers $p$ and $q$, can be constructed by Dehn surgery on an unknot with coefficient $-\frac{p}{q}$. One can instead construct $L(p,q)$ by an integral Dehn surgery on the framed ‘generalised Hopf link’
where the unknots have framings $a_{1}$, $\ldots$, $a_{n}$ from left to right, for a continued fraction representation of $-\frac{p}{q}$ as follows.
See also
Wikipedia, Lens space.
In the context of orbifolds:
Siddhartha Gadgil, Equivariant framings, lens spaces and contact structures, Pacific Journal of Mathematics, Vol. 208 (2003), No. 1, 73–84 (doi:10.2140/pjm.2003.208.73)
Michel Boileau, Steven Boyer, Radu Cebanu, Genevieve S. Walsh, Section 3 of: Knot commensurability and the Berge conjecture, Geom. Topol. 16 (2012) 625-664 (arXiv:1008.1034)
On the 3d-3d correspondence for lens spaces:
On classification of lens spaces:
Paul Olum, Mappings of manifolds and the notion of degree, Ann. of Math. (2) 58 (1953), p. 458–480. JSTOR:1969748
E. J. Brody, The topological classification of the lens spaces, Ann. of Math. (2) 71 (1960), 163–184. MR0116336 (22 #7125) Zbl 0119.18901
John Milnor, Whitehead torsion, Bull. Amer. Math. Soc. 72 (1966) 358–426 [doi:10:1090/S0002-9904-1966-11484-2, maths.ed.ac.uk, jstor:bams/1183527946]
Last revised on February 15, 2024 at 14:43:12. See the history of this page for a list of all contributions to it.