# 3-sphere

The **3-sphere** (written \( S^{3} \)) is the set of points in four-dimensional [Euclidean space](https://www.edgepedia.org/euclidean-space) at distance 1 from the origin, the three-dimensional analog of the familiar sphere surface. In coordinates it is the subspace of \( \mathbb{R}^{4} \) given by \( w^{2} + x^{2} + y^{2} + z^{2} = 1 \).<sup>[1](https://www.math.uni-bielefeld.de/~sek/top/hopf/hopf-fib.htm)</sup> It is the most basic example of a closed 3-manifold and plays a central role in topology, geometry, and physics.<sup>[2](https://math.arizona.edu/~glickenstein/rf/notes1.pdf)</sup>

| Key fact | Detail |
|---|---|
| Definition | Unit sphere in \( \mathbb{R}^{4} \): \( w^{2}+x^{2}+y^{2}+z^{2}=1 \)<sup>[1](https://www.math.uni-bielefeld.de/~sek/top/hopf/hopf-fib.htm)</sup> |
| Alternative model | One-point compactification of \( \mathbb{R}^{3} \); every loop contracts to a point, so \( S^{3} \) is simply connected<sup>[2](https://math.arizona.edu/~glickenstein/rf/notes1.pdf)</sup> |
| Group structure | Unit quaternions; isomorphic to \( SU(2) \), the double cover of \( SO(3) \)<sup>[2](https://math.arizona.edu/~glickenstein/rf/notes1.pdf)</sup> |
| Homogeneous space | \( SO(4)/SO(3) \), since rotations of \( \mathbb{R}^{4} \) act transitively with stabilizer \( SO(3) \)<sup>[2](https://math.arizona.edu/~glickenstein/rf/notes1.pdf)</sup> |
| Hyper-surface area | \( 2\pi^{2}R^{3} \) for radius \( R \)<sup>[3](https://mathworld.wolfram.com/Hypersphere.html)</sup> |
| Hopf fibration | Map \( h: S^{3} \to S^{2} \) (Heinz Hopf, 1931) whose fibers are circles, any two of which are linked<sup>[1](https://www.math.uni-bielefeld.de/~sek/top/hopf/hopf-fib.htm)</sup><sup> • </sup><sup>[4](https://people.math.harvard.edu/~knill/teaching/math22a2018/exhibits/threesphere/index.html)</sup> |
| Poincaré conjecture | \( S^{3} \) is the unique closed simply connected 3-manifold; proved by Grigoriy Perelman in 2002–2003 preprints using Ricci flow<sup>[5](https://www.claymath.org/millennium/poincare-conjecture/)</sup> |

## Naming conventions

The word "sphere" carries two incompatible conventions. Geometers count the dimension of the ambient space, so a geometer's "3-sphere" is the ordinary sphere surface in \( \mathbb{R}^{3} \), while topologists count the dimension of the object itself, so the topologist's \( S^{3} \) lives in \( \mathbb{R}^{4} \).<sup>[3](https://mathworld.wolfram.com/Hypersphere.html)</sup>

## Constructions and basic geometry

The 3-sphere can be constructed in several equivalent ways: as the set of unit vectors in \( \mathbb{R}^{4} \), or as the one-point compactification of \( \mathbb{R}^{3} \), the space obtained by adding a single point at infinity to ordinary three-dimensional space.<sup>[2](https://math.arizona.edu/~glickenstein/rf/notes1.pdf)</sup> Under the second description, any loop in \( S^{3} \) can be contracted to a point, which shows that \( S^{3} \) is simply connected.<sup>[2](https://math.arizona.edu/~glickenstein/rf/notes1.pdf)</sup>

The rotations \( SO(4) \) of \( \mathbb{R}^{4} \) act transitively on \( S^{3} \) with stabilizer \( SO(3) \), so \( S^{3} \) is the homogeneous space \( SO(4)/SO(3) \).<sup>[2](https://math.arizona.edu/~glickenstein/rf/notes1.pdf)</sup> Its hyper-surface area is \( 2\pi^{2}R^{3} \), obtained by integrating in the four-dimensional spherical coordinates \( x_{1} = R\sin\psi\sin\phi\cos\theta \), \( x_{2} = R\sin\psi\sin\phi\sin\theta \), \( x_{3} = R\sin\psi\cos\phi \), \( x_{4} = R\cos\psi \).<sup>[3](https://mathworld.wolfram.com/Hypersphere.html)</sup> Unlike the ordinary sphere, whose enclosed volume grows with radius without bound in the ambient space, the unit 3-sphere is itself a three-dimensional "universe": a compact space with no boundary in which a traveler moving straight ahead eventually returns to the start.

## Group structure: quaternions and SU(2)

The unit vectors of \( \mathbb{R}^{4} \) carry the group structure of the unit quaternions, and this group is isomorphic to \( SU(2) \), the double cover of the rotation group \( SO(3) \).<sup>[2](https://math.arizona.edu/~glickenstein/rf/notes1.pdf)</sup> The nLab describes the 3-sphere as the special unitary group \( S^{3} = SU(2) \), one of the spheres that are simultaneously groups (with \( S^{0} \) and \( S^{1} \)); the 7-sphere carries a related but weaker Moufang loop structure.<sup>[6](https://ncatlab.org/nlab/show/Hopf%20fibration)</sup> This identification underlies the [Hopf fibration](https://www.edgepedia.org/hopf-fibration)'s connection with rotations of 3-space, which is the basis for its applications to physics and engineering.<sup>[7](https://arxiv.org/abs/2212.01642)</sup>

## The Hopf fibration

The **Hopf fibration** is a map \( h: S^{3} \to S^{2} \), a fundamental example in differential geometry and algebraic topology, where \( S^{2} \) is the unit sphere in \( \mathbb{R}^{3} \).<sup>[1](https://www.math.uni-bielefeld.de/~sek/top/hopf/hopf-fib.htm)</sup> [Heinz Hopf](https://www.edgepedia.org/heinz-hopf) (1894–1971) studied the map in a 1931 paper; one explicit formula is

\[ f(x,y,z,w) = \left[ x^{2}+y^{2}-z^{2}-w^{2},\; 2(xw+yz),\; 2(yw-xz) \right]. \]

<sup>[4](https://people.math.harvard.edu/~knill/teaching/math22a2018/exhibits/threesphere/index.html)</sup> Using the identification of \( \mathbb{C} \cup \{\infty \} \) (the [Riemann sphere](https://www.edgepedia.org/riemann-sphere)) with \( S^{2} \) by stereographic projection, the map can be written compactly as \( h(w,x,y,z) = (w+ix)/(y+iz) \).<sup>[1](https://www.math.uni-bielefeld.de/~sek/top/hopf/hopf-fib.htm)</sup>

The defining property is that the preimage of each point of \( S^{2} \) is a circle in \( S^{3} \): the three-sphere is built from a collection of circles arranged like the points of a two-sphere.<sup>[8](https://ocw.mit.edu/courses/18-905-algebraic-topology-i-fall-2016/resources/18-905f16-th/)</sup> Two geometric facts make the construction remarkable. First, any two of these fiber circles are linked, and the circles must be lined up in a delicate way to ensure this.<sup>[1](https://www.math.uni-bielefeld.de/~sek/top/hopf/hopf-fib.htm)</sup> Second, the map generated the first calculated homotopy group \( \pi_{n}(S^{m}) \) with \( n > m \): the group \( \pi_{3}(S^{2}) \) is isomorphic to \( \mathbb{Z} \) and is generated by \( h \).<sup>[1](https://www.math.uni-bielefeld.de/~sek/top/hopf/hopf-fib.htm)</sup> The fibration is a canonical nontrivial principal \( U(1) \)-bundle over the 2-sphere with total space \( S^{3} \), and the Hopf construction produces four fibrations \( S^{0} \to S^{1} \), \( S^{1} \to S^{2} \), \( S^{3} \to S^{2} \), and \( S^{7} \to S^{4} \).<sup>[6](https://ncatlab.org/nlab/show/Hopf%20fibration)</sup>

## Role in three-manifold topology

**The Poincaré conjecture.** In 1904 [Henri Poincaré](https://www.edgepedia.org/henri-poincare) asked whether the three-dimensional sphere is characterized as the unique simply connected closed 3-manifold, knowing that the two-dimensional sphere is essentially characterized by simple connectivity.<sup>[5](https://www.claymath.org/millennium/poincare-conjecture/)</sup> The question resisted proof for nearly a century. Richard Hamilton introduced [Ricci flow](https://www.edgepedia.org/ricci-flow), the equation \( dg/dt = -2\,\mathrm{Ric} \), in 1982 and used it to prove that any closed 3-manifold with a metric of positive [Ricci curvature](https://www.edgepedia.org/ricci-curvature) admits a metric of constant positive sectional curvature, making it a quotient of the round 3-sphere by a finite group of isometries.<sup>[9](https://math.berkeley.edu/~lott/perelmanicm.pdf)</sup> In 2002 and 2003 Grigoriy Perelman posted preprints, building on Hamilton's program, that proved the Poincaré conjecture and, with it, William Thurston's geometrization conjecture, which says every closed 3-manifold is built from standard pieces each carrying one of eight well-understood geometries.<sup>[5](https://www.claymath.org/millennium/poincare-conjecture/)</sup> Perelman was offered the Fields Medal in 2006 and the Clay Millennium Prize, with its \$1 million award, in 2010; he declined both.<sup>[10](https://www.livescience.com/physics-mathematics/mathematics/science-history-russian-mathematician-quietly-publishes-paper-and-solves-one-of-the-most-famous-unsolved-conjectures-in-mathematics-nov-11-2002)</sup><sup> • </sup><sup>[11](https://www.claymath.org/library/annual_report/ar2010/ar2010.pdf)</sup>

**Spherical space forms.** Quotients of \( S^{3} \) by finite groups of isometries acting freely are called spherical space forms.<sup>[12](https://web.math.ucsb.edu/~cooper/37.pdf)</sup> The simplest are the lens spaces \( L(p,q) \), obtained as the orbit space of a cyclic group of order \( p \) acting freely on \( S^{3} \).<sup>[13](https://link.springer.com/article/10.1007/s00208-018-1792-7)</sup> Unlike the two-dimensional case, where the sphere and projective plane exhaust the spherical surfaces, there are infinitely many spherical 3-manifolds; every finite cyclic group occurs as the fundamental group of one, and the [Poincaré homology sphere](https://www.edgepedia.org/poincare-homology-sphere) is the quotient by the binary icosahedral group, a finite group of order 120.<sup>[14](https://imag.umontpellier.fr/~maillot/three_lectures.pdf)</sup> Lens spaces provide the known examples of distinct irreducible, atoroidal 3-manifolds sharing the same fundamental group.<sup>[12](https://web.math.ucsb.edu/~cooper/37.pdf)</sup>

**Recognition algorithms.** Deciding whether a given triangulated 3-manifold is the 3-sphere is a central algorithmic problem. Rubinstein (1995) and Thompson (1994) proved the problem decidable, using almost normal surfaces; Saul Schleimer showed in 2011 that it lies in the complexity class NP, and Marc Lackenby proved in 2021 that it also lies in co-NP.<sup>[15](https://link.springer.com/article/10.1007/s41468-022-00092-8)</sup> No polynomial-time algorithm is known, and the problem is a bottleneck shared with unknot recognition and prime decomposition.<sup>[16](https://arxiv.org/abs/1912.09051)</sup> Mijatović gave an explicit procedure that simplifies any triangulation of \( S^{3} \) with \( t \) tetrahedra to the canonical two-tetrahedron triangulation using fewer than \( a\,t^{2}2^{bt^{2}} \) Pachner moves, with \( a \le 6\cdot 10^{6} \) and \( b < 5\cdot 10^{4} \), yielding a new recognition algorithm.<sup>[17](https://arxiv.org/abs/math/0008107)</sup>

## Physics and applications

In general relativity, spatial \( S^{3} \)-topology plays a particularly important role in the standard model of cosmology based on the spatially homogeneous and isotropic Friedmann–Robertson–Walker solutions, and a spatial 3-sphere is a natural domain for numerical studies of Ricci flow and Maxwell theory.<sup>[18](https://www.sciencedirect.com/science/article/abs/pii/S0021999109003131)</sup>

In quantum information theory, the [Hilbert space](https://www.edgepedia.org/hilbert-space) of a single qubit is the 3-sphere \( S^{3} \): the \( S^{2} \) base of a suitably oriented Hopf fibration is the [Bloch sphere](https://www.edgepedia.org/bloch-sphere), while the circular fibers represent the qubit's overall phase degree of freedom.<sup>[19](https://arxiv.org/abs/quant-ph/0108137)</sup> For two qubits the Hilbert space is the 7-sphere \( S^{7} \), which admits a Hopf fibration with \( S^{3} \) fibers and an \( S^{4} \) base, and suitably oriented \( S^{7} \) fibrations are sensitive to entanglement.<sup>[19](https://arxiv.org/abs/quant-ph/0108137)</sup> More broadly, the Hopf fibration has physical applications including magnetic monopoles and rigid body mechanics.<sup>[7](https://arxiv.org/abs/2212.01642)</sup>

## References

1. [The Hopf fibration, University of Bielefeld](https://www.math.uni-bielefeld.de/~sek/top/hopf/hopf-fib.htm)
2. [Notes on spherical manifolds, G. Glickenstein, University of Arizona](https://math.arizona.edu/~glickenstein/rf/notes1.pdf)
3. [Hypersphere, Wolfram MathWorld](https://mathworld.wolfram.com/Hypersphere.html)
4. [Hopf fibration exhibit, Math 22a, Harvard University (O. Knill)](https://people.math.harvard.edu/~knill/teaching/math22a2018/exhibits/threesphere/index.html)
5. [Poincaré Conjecture, Clay Mathematics Institute](https://www.claymath.org/millennium/poincare-conjecture/)
6. [Hopf fibration, nLab](https://ncatlab.org/nlab/show/Hopf%20fibration)
7. [David W. Lyons, An Elementary Introduction to the Hopf Fibration, arXiv:2212.01642](https://arxiv.org/abs/2212.01642)
8. [18-905f16-th.gif, Algebraic Topology I, MIT OpenCourseWare](https://ocw.mit.edu/courses/18-905-algebraic-topology-i-fall-2016/resources/18-905f16-th/)
9. [John Lott, The work of Grigory Perelman, ICM 2006](https://math.berkeley.edu/~lott/perelmanicm.pdf)
10. [Science history: Russian mathematician quietly publishes paper, Live Science](https://www.livescience.com/physics-mathematics/mathematics/science-history-russian-mathematician-quietly-publishes-paper-and-solves-one-of-the-most-famous-unsolved-conjectures-in-mathematics-nov-11-2002)
11. [Clay Mathematics Institute Annual Report 2010](https://www.claymath.org/library/annual_report/ar2010/ar2010.pdf)
12. [Introduction to 3-manifolds (geometrization), University of California, Santa Barbara](https://web.math.ucsb.edu/~cooper/37.pdf)
13. [The isoperimetric problem for lens spaces, Mathematische Annalen, Springer](https://link.springer.com/article/10.1007/s00208-018-1792-7)
14. [Three Lectures on 3-manifolds, S. Maillot, Université de Montpellier](https://imag.umontpellier.fr/~maillot/three_lectures.pdf)
15. [Frontiers of sphere recognition in practice, Journal of Applied and Computational Topology, Springer](https://link.springer.com/article/10.1007/s41468-022-00092-8)
16. [On the hardness of finding normal surfaces, arXiv:1912.09051](https://arxiv.org/abs/1912.09051)
17. [Aleksandar Mijatović, Simplifying triangulations of S³, arXiv:math/0008107](https://arxiv.org/abs/math/0008107)
18. [A spectral solver for evolution problems with spatial S³-topology, Journal of Computational Physics, ScienceDirect](https://www.sciencedirect.com/science/article/abs/pii/S0021999109003131)
19. [R. Mosseri, R. Dandoloff, Geometry of entangled states, Bloch spheres and Hopf fibrations, arXiv:quant-ph/0108137](https://arxiv.org/abs/quant-ph/0108137)

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