nLab lens space




A lens space is the quotient space

L(n,{l i} i=0 d)S 2d+1/ nS( l 0 l d)/ n L\big(n,\{l_i\}_{i=0}^d\big) \;\coloneqq\; S^{2d+1} / \mathbb{Z}_n \;\coloneqq\; S \big( \mathbb{C}_{l_0} \oplus \cdots \oplus \mathbb{C}_{l_d} \big) / \mathbb{Z}_n

of a (2d+1)-dimensional sphere by a free action of a finite cyclic group n\mathbb{Z}_n; specifically — under the identification of S 2d+1S^{2d+1} with the unit sphere in the real vector space underlying d+1\mathbb{C}^{d+1} — by the action induced from a linear representation of the form

n× d+1 d+1 ([k],(z 0,,z d)) (exp(2πil 0n)z 0,,exp(2πil dn)z d) \array{ \mathbb{Z}_n \times \mathbb{C}^{d+1} & \overset{ }{ \longrightarrow } & \mathbb{C}^{d+1} \\ \big( [k], (z_0, \cdots, z_{d}) \big) &\mapsto& \Big( \exp( 2 \pi \mathrm{i} \tfrac{l_0}{n} ) \cdot z_0 , \cdots , \exp( 2 \pi \mathrm{i} \tfrac{l_d}{n} ) \cdot z_d \Big) }

for natural numbers l il_i coprime to nn: gcd(l i,n)=1gcd(l_i,n) = 1.

Often this is considered by default for 2d+1=32d + 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 3S^{3} and the product manifold S 2×S 1S^{2} \times S^{1} of the 2-sphere with the circle.




Two lens spaces L(n,{l i} i=0 d)L\big(n,\{l_i\}_{i=0}^d\big) and L(n,{l i} i=0 d)L\big(n,\{l'_i\}_{i=0}^d\big) are homotopy equivalent iff there exists a kk\in\mathbb{N} with

l 0l dk dl 0l dmodn. l_0\cdot\ldots\cdot l_d \;\equiv\; k^d l'_0 \cdot \ldots \cdot l'_d \mod n \,.

(Olum 1953)


Two lens spaces L(n,{l i} i=0 d)L\big(n,\{l_i\}_{i=0}^d\big) and L(n,{l i} i=0 d)L\big(n,\{l'_i\}_{i=0}^d\big) are homeomorphic iff there exists a kk\in\mathbb{N} and a permutation σSym d\sigma\in Sym_d with

l i±kl σ(i)modn l_i \;\equiv\; \pm k l'_{\sigma(i)} \mod n

for all i=0,,di=0,\ldots, d.

(Brody 1960, Milnor 1966)

In both theorems, the same nn for both lens spaces is a necessary condition for them to be homeomorphic or homotopy equivalent as follows with their respective fundamental group.

Homotopy theory

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.

Via Dehn surgery

The 3-dimensional lens space L(p,q)L(p,{1,q})L(p,q) \coloneqq L(p, \{1,q\}), for coprime integers pp and qq, can be constructed by Dehn surgery on an unknot with coefficient pq-\frac{p}{q}. One can instead construct L(p,q)L(p,q) by an integral Dehn surgery on the framed ‘generalised Hopf link’

where the unknots have framings a 1a_{1}, \ldots, a na_{n} from left to right, for a continued fraction representation of pq-\frac{p}{q} as follows.

a 11a 211a n a_{1} - \frac{1}{a_{2} - \frac{1}{\cdots - \frac{1}{a_{n}}}}


See also

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:

Last revised on February 15, 2024 at 14:43:12. See the history of this page for a list of all contributions to it.