(see also Chern-Weil theory, parameterized homotopy theory)
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
homotopy theory, (∞,1)-category theory, homotopy type theory
flavors: stable, equivariant, rational, p-adic, proper, geometric, cohesive, directed…
models: topological, simplicial, localic, …
see also algebraic topology
Introductions
Definitions
Paths and cylinders
Homotopy groups
Basic facts
Theorems
Let $X$ be an H-space. The Hopf construction (Hopf 35) on $X$ is a fibration
whose fiber is $X$, whose base space is the suspension of $X$, and whose total space is the join of $X$ with itself. (Stasheff 70, chapter 1).
Specialized to $X$ the sphere of dimension 0, 1, 3, or 7, the Hopf construction yields the Hopf fibrations. (And by the Hopf invariant one theorem these are the only dimensions for in which spheres are H-spaces.)
Write $I \coloneqq [0,1]$ for the unit interval, regarded as a topological space.
Let $X,Y$ be topological spaces.
The suspension $\Sigma X$ is the quotient space
by the equivalence relation given by
The join $X \ast Y$ is the quotient space
by the equivalence relation
Given a continuous function of the form
its Hopf construction is the continuous function
out of the join into the suspension, given in the coordinates of def. by
If $X$ is an “grouplike H-space”, in that it is a topological magma such that left multiplication acts by weak homotopy equivalences, then the homotopy fiber of the Hopf construction $X \ast X \to \Sigma X$ over any point is weakly homotopy equivalent to $X$ (here).
Beware that it may not generally be true that the ordinary fibers of the Hopf construction are weakly homotopy equivalent to the homotopy fibers, see also the discussion of quasifibrations below. But in the classical examples it Happens to be the case, see at Hopf fibration.
Consider $X = S^{n-1}$ a sphere
Given a continuous function
the degrees
are independent of the choice of $x \in S^{n-1}$. The Hopf invariant $h$ of the Hopf construction $H_f$ of $f$, def. , is the product of these two:
(Mosher-Tangora, exercises to section 4, page 38)
Beware that Stasheff 70, theorem 1.2 claims that Sugawara claimed that the Hopf construction for any CW H-space is necessarily a quasifibration. But it seems (here) that Sugawara never actually claimed this and also (here) that it is not actually the case.
A different but homotopy-equivalent realization of the Hopf construction, which over grouplike H-spaces is guaranteed to be a quasifibration, is maybe given in Dold-Lashof 59, see also Stasheff 70, theorem 1.4.
When $X$ is a sphere that is an $H$-space, namely, one of the groups $S^0 = \mathbb{Z}/2$ the group of order 2, $S^1 = U(1)$ the circle group, the 3-sphere special unitary group $S^3 = SU(2)$; or the 7-sphere $S^7$ with its Moufang loop structure, then the Hopf construction produces the four Hopf fibrations:
In detail, let $A \in \{\mathbb{R}, \mathbb{C}, \mathbb{H}, \mathbb{O}\}$ be one of the real normed division algebras and write
for its dimension as a real vector space. Then the $S^{n-1}$-sphere may be identified with the subspace of unit norm elements in $A$:
Consider then the pairing map
which is the restriction to these unit norm elements of the product in $A$:
This is well defined by the very property that for normed division algebras the norm is multiplicative.
Accordingly, the join of two such spheres is naturally parameterized as follows
which makes manifest that
Similarly, the suspension is parameterized by
where we take $I = [0,1]$ and $t \in I$. This makes manifest that
Moreover, in this parameterization the Hopf construction, def. , which is given by
manifestly gives the Hopf fibration map.
Notice that it is again the multiplicativity of the norm in division algebras which makes this work: if ${\vert x \vert}^2 = 2t$ and ${\vert y\vert}^2 = 2 - 2t$ then it follows that
hence that indeed we have a well-defined map like so:
The original sources are
Heinz Hopf, Über die Abbildungen von Sphären auf Sphären niedrigerer Dimension, Fund. Math. 25: 427–440 (1935) (Euclid)
George Whitehead, On the homotopy groups of spheres and rotation groups, Annals of Mathematics. Second Series 43 (4): 634–640, (1942) (JSTOR)
Albrecht Dold, Richard Lashof, Principal quasifibrations and fibre homotopy equivalence of bundles, Illinois J. Math. Volume 3, Issue 2 (1959), 285-305 (euclid:1255455128)
Review inclides
Textbook accounts include
Robert Mosher, Martin Tangora, p. 38 of Cohomology Operations and Application in Homotopy Theory, Harper and Row (1968) (pdf)
Marcelo Aguilar, Samuel Gitler, Carlos Prieto, section 10.6 of Algebraic topology from a homotopical viewpoint, Springer (2002) (toc pdf)
See also
Guillermo Moreno, Hopf construction map in higher dimensions (arXiv:math/0404172)
Feza Guersey, Chia-Hsiung Tze, (4b.2) in On the role of Division, Jordan and Related algebras in Particle Physics
Discussion of the situation in parameterized homotopy theory includes
A. L. Cook, M.C. Crabb, Fiberwise Hopf structures on sphere bundles, J. London Math. Soc. (2) 48 (1993) 365-384 (pdf)
Kouyemon Iriye, Equivariant Hopf structures on a sphere, J. Math. Kyoto Univ. Volume 35, Number 3 (1995), 403-412 (Euclid)
See also
Last revised on June 9, 2022 at 04:34:49. See the history of this page for a list of all contributions to it.