topology (point-set topology, point-free topology)
see also differential topology, algebraic topology, functional analysis and topological homotopy theory
Basic concepts
fiber space, space attachment
Extra stuff, structure, properties
Kolmogorov space, Hausdorff space, regular space, normal space
sequentially compact, countably compact, locally compact, sigma-compact, paracompact, countably paracompact, strongly compact
Examples
Basic statements
closed subspaces of compact Hausdorff spaces are equivalently compact subspaces
open subspaces of compact Hausdorff spaces are locally compact
compact spaces equivalently have converging subnet of every net
continuous metric space valued function on compact metric space is uniformly continuous
paracompact Hausdorff spaces equivalently admit subordinate partitions of unity
injective proper maps to locally compact spaces are equivalently the closed embeddings
locally compact and second-countable spaces are sigma-compact
Theorems
Analysis Theorems
manifolds and cobordisms
cobordism theory, Introduction
The $n$-dimensional unit sphere , or simply $n$-sphere, is the topological space given by the subset of the $(n+1)$-dimensional Cartesian space $\mathbb{R}^{n+1}$ consisting of all points $x$ whose distance from the origin is $1$
The $n$-dimensional sphere of radius $r$ is
Topologically, this is equivalent (homeomorphic) to the unit sphere for $r \gt 0$, or a point for $r = 0$.
This is naturally also a smooth manifold of dimension $n$, with the smooth structure induced from the standard sooth structure on $\mathbb{R}$^n.
One can also talk about a sphere in an arbitrary (possibly infinite-dimensional) normed vector space $V$:
If a locally convex topological vector space admits a continuous linear injection into a normed vector space, this can be used to define its sphere. If not, one can still define the sphere as a quotient of the space of non-zero vectors under the scalar action of $(0,\infty)$.
Homotopy theorists define $S^\infty$ to be the sphere in the (incomplete) normed vector space (traditionally with the $l^2$ norm) of infinite sequences almost all of whose values are $0$, which is the directed colimit of the $S^n$:
In themselves, these provide nothing new to homotopy theory, as they are at least weakly contractible and usually contractible. However, they are a very useful source of big contractible spaces and so are often used as a starting point for making concrete models of classifying spaces.
If the vector space is a shift space, then contractibility is straightforward to prove.
Let $V$ be a shift space of some order. Let $S V$ be its sphere (either via a norm or as the quotient of non-zero vectors). Then $S V$ is contractible.
Let $T \colon V \to V$ be a shift map. The idea is to homotop the sphere onto the image of $T$, and then down to a point.
It is simplest to start with the non-zero vectors, $V \setminus \{0\}$. As $T$ is injective, it restricts to a map from this space to itself which commutes with the scalar action of $(0,\infty)$. Define a homotopy $H \colon [0,1] \times (V \setminus \{0\}) \to V \setminus \{0\}$ by $H_t(v) = (1 - t)v + t T v$. It is clear that, assuming it is well-defined, it is a homotopy from the identity to $T$. To see that it is well-defined, we need to show that $H_t(v)$ is never zero. The only place where it could be zero would be on an eigenvector of $T$, but as $T$ is a shift map then it has none.
As $T$ is a shift map, it is not surjective and so we can pick some $v_0$ not in its image. Then we define a homotopy $G \colon [0,1] \times (V \setminus \{0\}) \to V \setminus \{0\}$ by $G_t(v) = (1 - t)T v + t v_0$. As $v_0$ is not in the image of $T$, this is well-defined on $V \setminus \{0\}$. Combining these two homotopies results in the desired contraction of $V \setminus \{0\}$.
If $V$ admits a suitable function defining a spherical subset (such as a norm) then we can modify the above to a contraction of the spherical subset simply by dividing out by this function. If not, as the homotopies above all commute with the scalar action of $(0,\infty)$, they descend to the definition of the sphere as the quotient of $V \setminus \{0\}$.
These spheres, or rather their underlying topological spaces or simplicial sets, are fundamental in (ungeneralised) homotopy theory. In a sense, Whitehead's theorem says that these are all that you need; no further generalised homotopy theory (in a sense dual to Eilenberg–Steenrod cohomology theory) is needed.
positive dimension spheres are H-cogroup objects, and this is the origin of the group structure on homotopy groups).
For $n \in \mathbb{N}$ the n-spheres are coset spaces of orthogonal groups
Similarly for the corresponding special orthogonal groups
and spin groups
and pin groups
Fix a unit vector in $\mathbb{R}^{n+1}$. Then its orbit under the defining $O(n+1)$-action on $\mathbb{R}^{n+1}$ is clearly the canonical embedding $S^n \hookrightarrow \mathbb{R}^{n+1}$. But precisely the subgroup of $O(n+1)$ that consists of rotations around the axis formed by that unit vector stabilizes it, and that subgroup is isomorphic to $O(n)$, hence $S^n \simeq O(n+1)/O(n)$.
Similarly, the analogous argument for unit spheres inside (the real vector spaces underlying) complex vector spaces, we have
For $k \in \mathbb{N}$ the (2k+1)-sphere $S^{2k+1}$ is the coset space of special unitary groups:
And still similarly, the analogous argument for unit spheres inside (the real vector spaces underlying) quaternionic vector spaces?, we have
For $k \in \mathbb{N}$, $k \geq 1$ the (4k-1)-sphere $S^{4k-1}$ is the coset space of quaternionic unitary groups:
Generally:
The connected Lie groups with effective transitive actions on n-spheres are precisely (up to isomorphism) the following:
with coset spaces
This goes back to Montogomery-Samelson 43, see
Gray-Green 70, p. 1-2
(also e.g Borel-Serre 53, 17.1)
The isomorphisms in Prop. and Prop. above hold in the category of topological spaces (homeomorphisms), but in fact also in the category of smooth manifolds (diffeomorphisms) and even in the category of Riemannian manifolds (isometries).
But the other coset space realizations of some n-spheres in Prop. are homeomorphisms, but not necessarily isometries (“squashed spheres”). There is also a double coset space realization which is not even a diffeomorphisms (“exotic sphere”, the Gromoll-Meyer sphere).
For more see 7-sphere – Coset space realization.
coset space-structures on n-spheres:
standard: | |
---|---|
$S^{n-1} \simeq_{diff} SO(n)/SO(n-1)$ | this Prop. |
$S^{2n-1} \simeq_{diff} SU(n)/SU(n-1)$ | this Prop. |
$S^{4n-1} \simeq_{diff} Sp(n)/Sp(n-1)$ | this Prop. |
exceptional: | |
$S^7 \simeq_{diff} Spin(7)/G_2$ | Spin(7)/G2 is the 7-sphere |
$S^7 \simeq_{diff} Spin(6)/SU(3)$ | since Spin(6) $\simeq$ SU(4) |
$S^7 \simeq_{diff} Spin(5)/SU(2)$ | since Sp(2) is Spin(5) and Sp(1) is SU(2), see Spin(5)/SU(2) is the 7-sphere |
$S^6 \simeq_{diff} G_2/SU(3)$ | G2/SU(3) is the 6-sphere |
$S^15 \simeq_{diff} Spin(9)/Spin(7)$ | Spin(9)/Spin(7) is the 15-sphere |
see also Spin(8)-subgroups and reductions
homotopy fibers of homotopy pullbacks of classifying spaces:
(from FSS 19, 3.4)
Precisely four spheres are parallelizable, and three of these are so via Lie group structure (hence are the only spheres with Lie group structure) (see at Hopf invariant one theorem):
$S^0$ (the group of order two, the group of units of the real numbers);
$S^1$ (the circle group, the group of unit complex numbers);
$S^3$ (the special unitary group $SU(2)$, the group of unit quaternions);
$S^7$ (the Moufang loop of unit octonions)
Every $n$-dimensional PL manifold admits a branched covering of the n-sphere (Alexander 20).
By the Riemann existence theorem, every connected compact Riemann surface admits the structure of a branched cover by a holomorphic function to the Riemann sphere. See there at branched cover of the Riemann sphere.
graphics grabbed from Chamseddine-Connes-Mukhanov 14, Figure 1, Connes 17, Figure 11
For 3-manifolds branched covering the 3-sphere see (Montesinos 74).
All PL 4-manifolds are simple branched covers of the 4-sphere (Piergallini 95, Iori-Piergallini 02).
But the n-torus for $n \geq 3$ is not a cyclic branched over of the n-sphere (Hirsch-Neumann 75)
(rational cohomology of iterated loop space of the 2k-sphere)
Let
(hence two positive natural numbers, one of them required to be even and the other required to be smaller than the first) and consider the D-fold loop space $\Omega^D S^n$ of the n-sphere.
Its rational cohomology ring is the free graded-commutative algebra over $\mathbb{Q}$ on one generator $e_{n-D}$ of degree $n - D$ and one generator $a_{2n - D - 1}$ of degree $2n - D - 1$:
(Kallel-Sjerve 99, Prop. 4.10)
The $(-1)$-sphere is the empty space.
The 0-sphere is the disjoint union of two points.
The 2-sphere is usual sphere from ordinary geometry. This canonically carries the structure of a complex manifold which makes it the Riemann sphere.
The 3-sphere and 4-sphere, 5-sphere and 6-sphere and 7-sphere are interesting, too.
The non-abelian generalized cohomology theory represented by n-spheres is Cohomotopy cohomology theory.
Axiomatization of the homotopy type of the 1-sphere (the circle) and the 2-sphere, as higher inductive types, is in
Visualization of the idea of the construction for the 2-sphere is in
Discussion of free group actions on spheres by finite groups includes
C. T. C. Wall, Free actions of finite groups on spheres, Proceedings of Symposia in Pure Mathematics, Volume 32, 1978 (pdf)
Alejandro Adem, Constructing and deconstructing group actions (arXiv:0212280)
The subgroups of SO(8) which act freely on $S^7$ have been classified in
and lifted to actions of Spin(8) in
Discussion of transitive actions on $n$-spheres by compact Lie groups:
Deane Montgomery, Hans Samelson, Transformation Groups of Spheres, Annals of Mathematics Second Series, Vol. 44, No. 3 (Jul., 1943), pp. 454-470 (jstor:1968975)
Alfred Gray, Paul S. Green, Sphere transitive structures and the triality automorphism, Pacific J. Math. Volume 34, Number 1 (1970), 83-96 (euclid:1102976640)
Further discussion of these actions is in
Paul de Medeiros, José Figueroa-O'Farrill, Sunil Gadhia, Elena Méndez-Escobar, Half-BPS quotients in M-theory: ADE with a twist, JHEP 0910:038,2009 (arXiv:0909.0163, pdf slides)
Paul de Medeiros, José Figueroa-O'Farrill, Half-BPS M2-brane orbifolds, Adv. Theor. Math. Phys. Volume 16, Number 5 (2012), 1349-1408. (arXiv:1007.4761, Euclid)
where they are related to the black M2-brane BPS-solutions of 11-dimensional supergravity at ADE-singularities.
See also the ADE classification of such actions on the 7-sphere (as discussed there)
Coset space structures on spheres:
The following to be handled with care:
The (isotopy class of an) embedding of a circle (1-sphere) into the 3-sphere is a knot. Discussion of embeddings of spheres of more general dimensions into each other:
Last revised on June 14, 2020 at 11:35:43. See the history of this page for a list of all contributions to it.