In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, in particular in
algebraic topology
Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism, though usually most classify ...
, the Hopf invariant is a
homotopy
In topology, a branch of mathematics, two continuous functions from one topological space to another are called homotopic (from grc, ὁμός "same, similar" and "place") if one can be "continuously deformed" into the other, such a deform ...
invariant of certain maps between
n-sphere
In mathematics, an -sphere or a hypersphere is a topological space that is homeomorphic to a ''standard'' -''sphere'', which is the set of points in -dimensional Euclidean space that are situated at a constant distance from a fixed point, call ...
s.
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Motivation
In 1931
Heinz Hopf
Heinz Hopf (19 November 1894 – 3 June 1971) was a German mathematician who worked on the fields of topology and geometry.
Early life and education
Hopf was born in Gräbschen, Germany (now , part of Wrocław, Poland), the son of Eliza ...
used
Clifford parallel In elliptic geometry, two lines are Clifford parallel or paratactic lines if the perpendicular distance between them is constant from point to point. The concept was first studied by William Kingdon Clifford in elliptic space and appears only in s ...
s to construct the ''
Hopf map''
:
,
and proved that
is essential, i.e., not
homotopic
In topology, a branch of mathematics, two continuous functions from one topological space to another are called homotopic (from grc, ὁμός "same, similar" and "place") if one can be "continuously deformed" into the other, such a deforma ...
to the constant map, by using the fact that the linking number of the circles
:
is equal to 1, for any
.
It was later shown that the
homotopy group
In mathematics, homotopy groups are used in algebraic topology to classify topological spaces. The first and simplest homotopy group is the fundamental group, denoted \pi_1(X), which records information about loops in a space. Intuitively, homotop ...
is the infinite
cyclic group
In group theory, a branch of abstract algebra in pure mathematics, a cyclic group or monogenous group is a group, denoted C''n'', that is generated by a single element. That is, it is a set of invertible elements with a single associative bina ...
generated by
. In 1951,
Jean-Pierre Serre
Jean-Pierre Serre (; born 15 September 1926) is a French mathematician who has made contributions to algebraic topology, algebraic geometry, and algebraic number theory. He was awarded the Fields Medal in 1954, the Wolf Prize in 2000 and the ina ...
proved that the
rational homotopy groups
:
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
In mathematics, a continuous function is a function such that a continuous variation (that is a change without jump) of the argument induces a continuous variation of the value of the function. This means that there are no abrupt changes in valu ...
(assume
). Then we can form the
cell complex
A CW complex (also called cellular complex or cell complex) is a kind of a topological space that is particularly important in algebraic topology. It was introduced by J. H. C. Whitehead (open access) to meet the needs of homotopy theory. This clas ...
:
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
In mathematics, a chain complex is an algebraic structure that consists of a sequence of abelian groups (or modules) and a sequence of homomorphisms between consecutive groups such that the image of each homomorphism is included in the kernel of t ...
, and since all boundary homomorphisms must be zero (recall that
), the cohomology is
:
Denote the generators of the cohomology groups by
:
and
For dimensional reasons, all cup-products between those classes must be trivial apart from
. Thus, as a ''ring'', the cohomology is
:
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