Homotopy invariant of maps between n-spheres
In mathematics, in particular in algebraic topology, the Hopf invariant is a homotopy invariant of certain maps between n-spheres.
Motivation
In 1931
Hopf map
![{\displaystyle \eta \colon S^{3}\to S^{2},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/81a37ef607170f32ec81b960624795da72001c65)
and proved that
is essential, i.e., not
homotopic
to the constant map, by using the fact that the linking number of the circles
![{\displaystyle \eta ^{-1}(x),\eta ^{-1}(y)\subset S^{3}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/adcc72d6c15c7b428000597e6b23f6421270d881)
is equal to 1, for any
.
It was later shown that the homotopy group
is the infinite cyclic group generated by
. In 1951,
rational homotopy groups
[1]
![{\displaystyle \pi _{i}(S^{n})\otimes \mathbb {Q} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/a9db5d3ed62131c3c5dc79274db0678fdcc142c0)
for an odd-dimensional sphere (
odd) are zero unless
is equal to 0 or n. However, for an even-dimensional sphere (n even), there is one more bit of infinite cyclic homotopy in degree
.
Definition
Let
be a
continuous map
(assume
![{\displaystyle n>1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ee74e1cc07e7041edf0fcbd4481f5cd32ad17b64)
). Then we can form the
cell complex
![{\displaystyle C_{\varphi }=S^{n}\cup _{\varphi }D^{2n},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/db36bdb47dc53e33ee8e60d56cef8435aa07c0c6)
where
is a
-dimensional disc attached to
via
.
The cellular chain groups
are just freely generated on the
-cells in degree
, so they are
in degree 0,
and
and zero everywhere else. Cellular (co-)homology is the (co-)homology of this chain complex, and since all boundary homomorphisms must be zero (recall that
), the cohomology is
![{\displaystyle H_{\mathrm {cell} }^{i}(C_{\varphi })={\begin{cases}\mathbb {Z} &i=0,n,2n,\\0&{\text{otherwise}}.\end{cases}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/020cad9c0921296931e71eba547cef1c93f661fc)
Denote the generators of the cohomology groups by
and ![{\displaystyle H^{2n}(C_{\varphi })=\langle \beta \rangle .}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0e98812c4eff2e18d1ead32441b52cabf673ca68)
For dimensional reasons, all cup-products between those classes must be trivial apart from
. Thus, as a ring, the cohomology is
![{\displaystyle H^{*}(C_{\varphi })=\mathbb {Z} [\alpha ,\beta ]/\langle \beta \smile \beta =\alpha \smile \beta =0,\alpha \smile \alpha =h(\varphi )\beta \rangle .}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ab6ae6dd82a0f534e143d89c835d2ff9d884a756)
The integer
is the Hopf invariant of the map
.
Properties
Theorem: The map
is a homomorphism.
If
is odd,
is trivial (since
is torsion).
If
is even, the image of
contains
. Moreover, the image of the Whitehead product of identity maps equals 2, i. e.
, where
is the identity map and
is the Whitehead product.
The Hopf invariant is
for the Hopf maps, where
, corresponding to the real division algebras
, respectively, and to the fibration
sending a direction on the sphere to the subspace it spans. It is a theorem, proved first by Frank Adams, and subsequently by Adams and Michael Atiyah with methods of topological K-theory, that these are the only maps with Hopf invariant 1.
Whitehead integral formula
J. H. C. Whitehead has proposed the following integral formula for the Hopf invariant.[2][3]: prop. 17.22
Given a map
, one considers a volume form
on
such that
.
Since
, the pullback
is a
closed differential form
:
![{\displaystyle d(\varphi ^{*}\omega _{n})=\varphi ^{*}(d\omega _{n})=\varphi ^{*}0=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/29c233b2c84cd7a022808f6168884353211cb46c)
.
By
exact differential form
: there exists an
![{\displaystyle (n-1)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/df88c6333caaf6471cf277f24b802ff9931b133e)
-form
![{\displaystyle \eta }](https://wikimedia.org/api/rest_v1/media/math/render/svg/e4d701857cf5fbec133eebaf94deadf722537f64)
on
![{\displaystyle S^{2n-1}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/171b6c25cdf8ddc2aeba5c1c2b5453ea561ac261)
such that
![{\displaystyle d\eta =\varphi ^{*}\omega _{n}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a770c5134316bf05d713757a1df399e818a5806f)
. The Hopf invariant is then given by
![{\displaystyle \int _{S^{2n-1}}\eta \wedge d\eta .}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7d324ba2041af32dd8c6a68a249be6ce5f2cebad)
Generalisations for stable maps
A very general notion of the Hopf invariant can be defined, but it requires a certain amount of homotopy theoretic groundwork:
Let
denote a vector space and
its
one-point compactification
, i.e.
![{\displaystyle V\cong \mathbb {R} ^{k}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/31c1aa732742781f8234cf61aa546d1a0baea0ea)
and
for some
.
If
is any pointed space (as it is implicitly in the previous section), and if we take the point at infinity to be the basepoint of
, then we can form the wedge products
![{\displaystyle V^{\infty }\wedge X.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0dfdb67e8dafa7efca8fa5403c87cfe9a91c850c)
Now let
![{\displaystyle F\colon V^{\infty }\wedge X\to V^{\infty }\wedge Y}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9b5f34195f3968baee63217c9f3d674fc49ba382)
be a stable map, i.e. stable under the
reduced suspension
functor. The
(stable) geometric Hopf invariant of
![{\displaystyle F}](https://wikimedia.org/api/rest_v1/media/math/render/svg/545fd099af8541605f7ee55f08225526be88ce57)
is
![{\displaystyle h(F)\in \{X,Y\wedge Y\}_{\mathbb {Z} _{2}},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/314b4171dd0703f37bda7e8b16727a0705856552)
an element of the stable
-equivariant homotopy group of maps from
to
. Here "stable" means "stable under suspension", i.e. the direct limit over
(or
, if you will) of the ordinary, equivariant homotopy groups; and the
-action is the trivial action on
and the flipping of the two factors on
. If we let
![{\displaystyle \Delta _{X}\colon X\to X\wedge X}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d64cf0a75108d2e011ff4397a84fb7f93e3069cc)
denote the canonical diagonal map and
the identity, then the Hopf invariant is defined by the following:
![{\displaystyle h(F):=(F\wedge F)(I\wedge \Delta _{X})-(I\wedge \Delta _{Y})(I\wedge F).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6aea41ab412f3ba89d1f8b56a0962e87a017d642)
This map is initially a map from
to ![{\displaystyle V^{\infty }\wedge V^{\infty }\wedge Y\wedge Y,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a72e642e3535436890534cefda188d4b2be58f36)
but under the direct limit it becomes the advertised element of the stable homotopy
-equivariant group of maps.
There exists also an unstable version of the Hopf invariant
, for which one must keep track of the vector space
.
References