Editing Homotopy (section)

==Invariance==
Homotopy equivalence is important because in [[algebraic topology]] many concepts are '''homotopy invariant''', that is, they respect the relation of homotopy equivalence.  For example, if ''X'' and ''Y'' are homotopy equivalent spaces, then:
* ''X'' is [[connected space|path-connected]] if and only if ''Y'' is.
* ''X'' is [[simply connected]] if and only if ''Y'' is.
* The (singular) [[homology (mathematics)|homology]] and [[cohomology group]]s of ''X'' and ''Y'' are [[group isomorphism|isomorphic]].
* If ''X'' and ''Y'' are path-connected, then the [[fundamental group]]s of ''X'' and ''Y'' are isomorphic, and so are the higher [[homotopy group]]s. (Without the path-connectedness assumption, one has π<sub>1</sub>(''X'',&thinsp;''x''<sub>0</sub>) isomorphic to π<sub>1</sub>(''Y'', ''f''(''x''<sub>0</sub>)) where {{nowrap|1=''f'' : ''X'' → ''Y''}} is a homotopy equivalence and {{nowrap|1=''x''<sub>0</sub> &isin; ''X''.)}}

An example of an algebraic invariant of topological spaces which is not homotopy-invariant is [[compactly supported homology]] (which is, roughly speaking, the homology of the [[compactification (mathematics)|compactification]], and compactification is not homotopy-invariant).