Difference between revisions of "Applying pathconnectedness"
m (2 revisions) 

(No difference)

Latest revision as of 19:31, 11 May 2008
This is a survey article about applying the concept/definition/theorem: pathconnectedness
This article is about how one can use a point settopological fact that a topological space (possibly with a lot of additional structure) is pathconnected.
The hypothesis of being pathconnected is in general stronger than the hypothesis of being connected, although for locally pathconnected spaces, such as manifolds, the two hypotheses are equivalent. In certain situations, connectedness is a more useful hypothesis, while in other situations, it is more useful to use pathconnectedness. Refer the article on applying connectedness.
Contents
Constructing paths and homotopies
The most useful and direct application of pathconnectedness is to construct paths, which can be used to define homotopies and change basepoints.
Basepointindependence of homotopy groups
The homotopy groups of a topological space are usually defined with respect to a basepoint; in order to prove that the homotopy groups at two basepoints are isomorphic, we use a path between them to give an isomorphism. In particular, the fundamental groups at any two basepoints in a pathconnected space are the same. Moreover, if is a pathconnected space, and is a space with nondegenerate basepoint , then the homotopy classes of based maps from to can be identified for different choices of basepoint .
Further information: Actions of the fundamental group
Multiplication maps in a group
In a pathconnected topological group, the multiplication map by any group element is homotopyequivalent to the identity map. The homotopy in this case is given by the path joining the group element to the identity element. In particular, any map given by an algebraic formula is homotopyequivalent to a power map.
This fact is useful, for instance, in showing that any compact connected nontrivial Lie group has zero Euler characteristic. The idea is that the fixedpoint free map of left multiplication by a group element, is homotopyequivalent to the identity map.
Constructing more complicated homotopies
A homotopy can be viewed as a kind of generalized path, and we can often use a path, or a loop, as the starting point of a homotopy. The typical additional concepts we need are those of a cofibration, and we often need to use tricks such as the two sides lemma or three sides lemma.