The integer
where
is the number of
-dimensional
cells in
.
It was given this name in honour of
L. Euler,
who proved in
1758
that the number
of vertices, the number
of edges and the number
of faces of a convex polyhedron are connected by the formula
.
This relation was known, in an implicit form, already
to
R. Descartes
(
1620).
It turns out that
where
is the
-dimensional
Betti number
of the complex
(the
Euler–Poincaré formula).
The Euler characteristic of
is a homology, homotopy and topological invariant of
.
In particular, it does not depend on the way
in which the space is partitioned into cells. Consequently one
can speak, for example, of the Euler characteristic of an
arbitrary compact polyhedron, meaning by it the Euler characteristic of
any of its triangulations. On the other hand, the Euler–Poincaré
formula permits the extension of the concept of the Euler characteristic
to a larger class of spaces and pairs of spaces for
which the right-hand side of the formula remains meaningful. This formula
has been generalized to the case of an arbitrary field
,
where it expresses the Euler characteristic in terms of the dimensions over
of the homology groups with coefficients in
:
Let
be a locally trivial fibration with fibre
.
If the spaces
,
and
satisfy certain conditions, then their Euler
characteristics are connected by the relation
.
In particular, the Euler characteristic of the direct product of
two spaces is equal to the product of their Euler characteristics. The relation
,
which holds for any excisive triple
,
makes it possible to compute the Euler characteristics of
all compact two-dimensional manifolds. The Euler characteristic of a sphere with
handles and
deleted open discs is
,
while that of a sphere with
Möbius strips and
deleted discs is
.
The Euler characteristic of an arbitrary compact orientable manifold of odd
dimension is equal to half that of its boundary. In particular,
the Euler characteristic of a closed orientable manifold of
odd dimension is zero, since its boundary is empty.