# Multiplications on Spheres (Jie Wu)

The **n**-dimensional sphere S^{n} is the set of unit
vectors in the **(n+1)**-dimensional Euclidean space
R^{n+1}. As a (topological) space, S^{n} is regarded as a
subspace of R^{n+1}.
For instance, S^{0}={-1,1} consists of two points, S^{1}
is the circle and S^{2} is the usual sphere.

200 years ago people developed group theory in mathematics. Roughly
speaking, a group means a set in which there is a binary operation
(so-called multiplication) with the properties 1) there is an identity
element, that is, there is an element, denoted by 1, such that
*1x=x1=x*; 2) this binary operation is associative, that is,
*(ab)c=a(bc)* and 3) the inverse exists, that is, for each element
*x*, there is another element, denoted by *x*^{-1}, such
that *xx*^{-1}=x^{-1}x=1.

A topological group **G** means a (topological) space **G** with
a ** continuous** multiplication on **G** such that **G** is
group under this multiplication and the function which sends *x* to
*x*^{-1} is continuous. For instance, the space of orthogonal
matrices is a topological group, where the multiplication is given by the
matrix multiplication. The Euclidean space is a topological group under
the vector addition.

An H-space (Hopf space) means a space **X** with a **
continuous**
multiplication on **X** such that this multiplication has the identity
element. In
other words, for an H-space, the multiplication need NOT be associative
and we do not require that the inverse exists too. A topological group is
certainly an H-space, but conversely it may not be true in general.

**Question 1.** Is S^{n} a topological
group?

When *n=0,1,3*, the answer is YES. The multiplication on
S^{1} and S^{3} are induced by the product of complex
numbers and quaternion numbers, respectively, where we regard complex
numbers as 2-dimnesional vectors and quaternion numbers as 4-dimnesional
vectors. For instance, when *n=1*, first we need to identify
S^{1} with complex numbers of length 1. Then we use the product of
two complex numbers to define the multiplication on S^{1}. The
point here is that the product of two complex numbers of length 1 is still
of length 1.
The multiplication on S^{3} follows from the same ideas, but you
may have to read some references on quaternion numbers to see how to
define product of 4-dimensional vectors.

** Question 2.** Is S^{n} an H-space?

In addition to the cases above, the answer is YES when *n=7*. The
multiplication on S^{7} is induced by the product of Cayley
numbers. Again you may have to read references on Cayley numbers to see
how to define product of 8-dimensional vectors.

The questions above are called Hopf invariant one problem. Adams solved
this problem in 1950s in his famous paper. Homotopy theory was shown her
powers in solving this problem. Adams' answer is as follows.

** Theorem 1:** S^{n} is a topological group if and
only if *n=0,1,3*.

** Theorem 2:** S^{n} is an H-space if and only if
*n=0,1,3,7*
Theorem 2 tells that if *n* is not equal to *0,1,3,7*, there is
no way to make a ** continuous** multiplication on S^{n} such
that S^{n} has the identity element under this multiplication. As
a good exercise, you may try to look at why we could not make a continuous
multiplication on the sphere S^{2}. You may regard S^{2}
as adding the infinite point to complex numbers. For sure we can define a
multiplication on complex numbers. Say vector addition. However whatever
multiplication defined on complex numbers is impossible continuously
extended to S^{2}.
Theorem 1 tells that S^{7} is never a topological group under
whatever multiplication. In fact, whatever multiplication defined on
S^{7} with identity is never associative. Homotopy theory also
tells that whatever multiplication defined on S^{3} with identity
is never commutative.
Now you may want to know how to prove the theorems above. Adams' ideas are
as follows: By assuming that S^{n} admits a multiplication with
identity, one gets certain "operations" on so-called homology
of certain space. After a long discussion, Adams obtained a contradiction
when n is not equals to 0,1,3,7. Thus he proved Theorem 2.
For Theorem 1, homotopy theory shows that any topological group admits a
so-called classifying space. By assuming that S^{7} adimits a
topological group structure, one gets a classifying space for
S^{7}. Again homotopy theory shows that this is impossible by
considering so-called the cohomology of the resulting classifying space.