nLab lens space

Idea

A lens space is the quotient space

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

of a (2d+1)-dimensional sphere by a free action of a finite cyclic group $\mathbb{Z}_n$; 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

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

for natural numbers $l_i$ 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.

Properties

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 $L(p,q) \coloneqq L(p, \{1,q\})$, 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.

$$a_{1} - \frac{1}{a_{2} - \frac{1}{\cdots - \frac{1}{a_{n}}}}$$

Related concepts

• Seifert 3-manifold

References

See also

• Manifold Atlas, Lens spaces.

• 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:

• Du Pei, Chapter III of: 3d-3d correspondence for Seifert manifolds, 2016 (spire:1469350, pdf)