Hopf invariant#Properties

In 1931 Heinz Hopf used Clifford parallels to construct the Hopf map

:$\backslash eta\backslash colon\; S^3\; \backslash to\; S^2,$

and proved that $\backslash eta$ is essential, i.e., not homotopic to the constant map, by using the fact that the linking number of the circles

:$\backslash eta^\{-1\}(x),\backslash eta^\{-1\}(y)\; \backslash subset\; S^3$

is equal to 1, for any $x\; \backslash neq\; y\; \backslash in\; S^2$.

It was later shown that the homotopy group $\backslash pi\_3(S^2)$ is the infinite cyclic group generated by $\backslash eta$. In 1951, Jean-Pierre Serre proved that the rational homotopy groups {{cite journal |last1=Serre |first1=Jean-Pierre |title=Groupes D'Homotopie Et Classes De Groupes Abeliens |journal=The Annals of Mathematics |date=September 1953 |volume=58 |issue=2 |pages=258–294 |doi=10.2307/1969789|jstor=1969789 }}

:$\backslash pi\_i(S^n)\; \backslash otimes\; \backslash mathbb\{Q\}$

for an odd-dimensional sphere ($n$ odd) are zero unless $i$ 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 $2n-1$.

Let $\backslash varphi\; \backslash colon\; S^\{2n-1\}\; \backslash to\; S^n$ be a continuous map (assume $n>1$). Then we can form the cell complex

: $C\_\backslash varphi\; =\; S^n\; \backslash cup\_\backslash varphi\; D^\{2n\},$

where $D^\{2n\}$ is a $2n$-dimensional disc attached to $S^n$ via $\backslash varphi$.

The cellular chain groups $C^*\_\backslash mathrm\{cell\}(C\_\backslash varphi)$ are just freely generated on the $i$-cells in degree $i$, so they are $\backslash mathbb\{Z\}$ in degree 0, $n$ and $2n$ 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 $n>1$), the cohomology is

:

Denote the generators of the cohomology groups by

: $H^n(C\_\backslash varphi)\; =\; \backslash langle\backslash alpha\backslash rangle$ and $H^\{2n\}(C\_\backslash varphi)\; =\; \backslash langle\backslash beta\backslash rangle.$

For dimensional reasons, all cup-products between those classes must be trivial apart from $\backslash alpha\; \backslash smile\; \backslash alpha$. Thus, as a ring, the cohomology is

: $H^*(C\_\backslash varphi)\; =\; \backslash mathbb\{Z\}[\backslash alpha,\backslash beta]/\backslash langle\; \backslash beta\backslash smile\backslash beta\; =\; \backslash alpha\backslash smile\backslash beta\; =\; 0,\; \backslash alpha\backslash smile\backslash alpha=h(\backslash varphi)\backslash beta\backslash rangle.$

The integer $h(\backslash varphi)$ is the Hopf invariant of the map $\backslash varphi$.

Theorem: The map $h\backslash colon\backslash pi\_\{2n-1\}(S^n)\backslash to\backslash mathbb\{Z\}$ is a homomorphism.

If $n$ is odd, $h$ is trivial (since $\backslash pi\_\{2n-1\}(S^n)$ is torsion).

If $n$ is even, the image of $h$ contains $2\backslash mathbb\{Z\}$. Moreover, the image of the Whitehead product of identity maps equals 2, i. e. $h([i\_n,\; i\_n])=2$, where $i\_n\; \backslash colon\; S^n\; \backslash to\; S^n$ is the identity map and $[\backslash ,\backslash cdot\backslash ,,\backslash ,\backslash cdot\backslash ,]$ is the Whitehead product.

The Hopf invariant is $1$ for the Hopf maps, where $n=1,2,4,8$, corresponding to the real division algebras $\backslash mathbb\{A\}=\backslash mathbb\{R\},\backslash mathbb\{C\},\backslash mathbb\{H\},\backslash mathbb\{O\}$, respectively, and to the fibration $S(\backslash mathbb\{A\}^2)\backslash to\backslash mathbb\{PA\}^1$ 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.

J. H. C. Whitehead has proposed the following integral formula for the Hopf invariant.{{cite journal |last1=Whitehead |first1=J. H. C. |title=An Expression of Hopf's Invariant as an Integral |journal=Proceedings of the National Academy of Sciences |date=1 May 1947 |volume=33 |issue=5 |pages=117–123 |doi=10.1073/pnas.33.5.117|pmid=16578254 |doi-access=free |pmc=1079004 |bibcode=1947PNAS...33..117W }}{{cite book |last1=Bott |first1=Raoul |last2=Tu |first2=Loring W |title=Differential forms in algebraic topology |date=1982 |location=New York |isbn=9780387906133}}{{rp|prop. 17.22}}

Given a map $\backslash varphi\; \backslash colon\; S^\{2n-1\}\; \backslash to\; S^n$, one considers a volume form $\backslash omega\_n$ on $S^n$ such that $\backslash int\_\{S^n\}\backslash omega\_n\; =\; 1$.

Since $d\backslash omega\_n\; =\; 0$, the pullback $\backslash varphi^*\; \backslash omega\_n$ is a closed differential form: $d(\backslash varphi^*\; \backslash omega\_n)\; =\; \backslash varphi^*\; (d\backslash omega\_n)\; =\; \backslash varphi^*\; 0\; =\; 0$.

By Poincaré's lemma it is an exact differential form: there exists an $(n\; -\; 1)$-form $\backslash eta$ on $S^\{2n\; -\; 1\}$ such that $d\backslash eta\; =\; \backslash varphi^*\; \backslash omega\_n$. The Hopf invariant is then given by

:$$

\int_{S^{2n - 1}} \eta \wedge d \eta.

A very general notion of the Hopf invariant can be defined, but it requires a certain amount of homotopy theoretic groundwork:

Let $V$ denote a vector space and $V^\backslash infty$ its one-point compactification, i.e. $V\; \backslash cong\; \backslash mathbb\{R\}^k$ and

:$V^\backslash infty\; \backslash cong\; S^k$ for some $k$.

If $(X,x\_0)$ 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 $V^\backslash infty$, then we can form the wedge products

:$V^\backslash infty\; \backslash wedge\; X.$

Now let

:$F\; \backslash colon\; V^\backslash infty\; \backslash wedge\; X\; \backslash to\; V^\backslash infty\; \backslash wedge\; Y$

be a stable map, i.e. stable under the reduced suspension functor. The (stable) geometric Hopf invariant of $F$ is

:$h(F)\; \backslash in\; \backslash \{X,\; Y\; \backslash wedge\; Y\backslash \}\_\{\backslash mathbb\{Z\}\_2\},$

an element of the stable $\backslash mathbb\{Z\}\_2$-equivariant homotopy group of maps from $X$ to $Y\; \backslash wedge\; Y$. Here "stable" means "stable under suspension", i.e. the direct limit over $V$ (or $k$, if you will) of the ordinary, equivariant homotopy groups; and the $\backslash mathbb\{Z\}\_2$-action is the trivial action on $X$ and the flipping of the two factors on $Y\; \backslash wedge\; Y$. If we let

:$\backslash Delta\_X\; \backslash colon\; X\; \backslash to\; X\; \backslash wedge\; X$

denote the canonical diagonal map and $I$ the identity, then the Hopf invariant is defined by the following:

:$h(F)\; :=\; (F\; \backslash wedge\; F)\; (I\; \backslash wedge\; \backslash Delta\_X)\; -\; (I\; \backslash wedge\; \backslash Delta\_Y)\; (I\; \backslash wedge\; F).$

This map is initially a map from

:$V^\backslash infty\; \backslash wedge\; V^\backslash infty\; \backslash wedge\; X$ to $V^\backslash infty\; \backslash wedge\; V^\backslash infty\; \backslash wedge\; Y\; \backslash wedge\; Y,$

but under the direct limit it becomes the advertised element of the stable homotopy $\backslash mathbb\{Z\}\_2$-equivariant group of maps.

There exists also an unstable version of the Hopf invariant $h\_V(F)$, for which one must keep track of the vector space $V$.

{{Reflist}}

• {{citation

| first = J. Frank|last= Adams|authorlink=Frank Adams

| year = 1960

| title = On the non-existence of elements of Hopf invariant one

| journal = Annals of Mathematics

| volume = 72

| pages = 20–104

| doi = 10.2307/1970147

| issue = 1

| jstor = 1970147

| mr=0141119

|citeseerx= 10.1.1.299.4490}}

• {{citation

| first1 = J. Frank|last1= Adams| author1-link=Frank Adams

| first2 = Michael F.|last2=Atiyah| author2-link=Michael Atiyah

| year = 1966

| title = K-Theory and the Hopf Invariant

| journal = Quarterly Journal of Mathematics

| volume = 17

| issue = 1

| pages = 31–38

| doi = 10.1093/qmath/17.1.31

| mr=0198460

}}

• {{cite web

| first1 = Michael|last1= Crabb

| first2=Andrew |last2= Ranicki|author2-link=Andrew Ranicki

| year = 2006

| title = The geometric Hopf invariant

| url = http://www.maths.ed.ac.uk/~aar/slides/hopfbeam.pdf

}}

• {{Citation | last1=Hopf | first1=Heinz | title=Über die Abbildungen der dreidimensionalen Sphäre auf die Kugelfläche | year=1931 | journal=Mathematische Annalen | issn=0025-5831 | volume=104 | pages=637–665 | doi=10.1007/BF01457962}}
• {{springer|first=A.V. |last=Shokurov|title=Hopf invariant|id=h/h048000}}

Category:Homotopy theory