Fundamental group
Poincaré group
The first absolute homotopy group . Let
be the interval
, and let
be its boundary. The elements of the fundamental group of the pointed topological space
are the homotopy classes of closed paths in
, that is, homotopy classes
of continuous mappings of the pair
into
. The path
:
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is called the product of and
. The homotopy class of the product depends only on the classes of the factors, and the resulting operation is, generally speaking, non-commutative. The identity is the class of the constant mapping into
, and the inverse of the class
containing the path
is the class of the path
. To a continuous mapping
corresponds the homomorphism
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that is, is a functor from the category of pointed topological spaces into the category of (non-Abelian) groups. For any path
joining the points
and
, one can define an isomorphism
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that depends only on the homotopy class of . The group
acts as a group of automorphisms on
, and in the case
,
acts as an inner automorphism
. The Hurewicz homomorphism
is an epimorphism with kernel
(Poincaré's theorem).
A path-connected topological space with a trivial fundamental group is called simply connected. The fundamental group of a product of spaces is isomorphic to the direct product of the fundamental groups of the factors:
. Let
be a path-connected topological space, and let
be a covering of
by a system of open sets
, closed under intersection, such that
; then
is the direct limit of the diagram
, where
, and
is induced by the inclusion
(the Seifert–van Kampen theorem). For example, if one is given a covering consisting of
,
and
, and if
is simply connected, then
is the free product of
and
. In the case of a CW-complex, the assertion of the theorem is also true for closed CW-subspaces of
.
For a CW-complex whose zero-dimensional skeleton consists of a single point
, each one-dimensional cell
gives a generator of
, and each two-dimensional cell
gives a relation corresponding to the attaching mapping of
.
Suppose that has a covering
such that the inclusion homomorphism
is zero for every point
. Then there is a covering
with
. In this case the group of homeomorphisms of
onto itself that commute with
(covering transformations) is isomorphic to
, and the order of
is equal to the cardinality of the fibre
. For a mapping
of path-connected spaces such that
there is a lifting
,
. The covering
is called universal.
References
[1] | W.S. Massey, "Algebraic topology: an introduction" , Springer (1977) |
[2] | V.A. Rokhlin, D.B. Fuks, "Beginner's course in topology. Geometric chapters" , Springer (1984) (Translated from Russian) |
[3] | E.H. Spanier, "Algebraic topology" , McGraw-Hill (1966) |
[4] | J.R. Stallings, "Group theory and three-dimensional manifolds" , Yale Univ. Press (1972) |
Comments
References
[a1] | B. Gran, "Homology theory" , Acad. Press (1975) |
Fundamental group. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Fundamental_group&oldid=47025