Section 51: Homotopy of Paths
Two continuous funcitons
and
from
to
are called homotopic if there is a continuous map
(
) such that
and
for all
.
itself is called a homotopy between
and
.
- Notation: .
- is an equivalence relation.
- If is constant, is called nulhomotopic.
- A space is said to be contractible if the identity map is nulhomotopic.
Two paths
and
in
(with the same domain
) are called path homotopic if there is a homotopy
between
and
such that
and
are constant.
- Notation: .
- is an equivalence relation.
- represents the path-homotopy equivalence class of .
If
and
are two paths in
such that
, then the product
of
and
is the joint path from
to
.
- is well-defined.
- satisfies the groupoid properties: associativity, when applied, the existence of the left and right identities, and the existence of the inverse using the reverse of : .