Examples
One important use of relative homology is the computation of the homology groups of quotient spaces . In the case that is a subspace of fulfilling the mild regularity condition that there exists a neighborhood of that has as a deformation retract, then the group is isomorphic to . We can immediately use this fact to compute the homology of a sphere. We can realize as the quotient of an n-disk by its boundary, i.e. . Applying the exact sequence of relative homology gives the following:
Because the disk is contractible, we know its homology groups vanish in all dimensions, so the above sequence collapses to the short exact sequence:
Therefore, we get isomorphisms . We can now proceed by induction to show that . Now because is the deformation retract of a suitable neighborhood of, we get that
Read more about this topic: Relative Homology
Famous quotes containing the word examples:
“In the examples that I here bring in of what I have [read], heard, done or said, I have refrained from daring to alter even the smallest and most indifferent circumstances. My conscience falsifies not an iota; for my knowledge I cannot answer.”
—Michel de Montaigne (1533–1592)
“It is hardly to be believed how spiritual reflections when mixed with a little physics can hold people’s attention and give them a livelier idea of God than do the often ill-applied examples of his wrath.”
—G.C. (Georg Christoph)
“No rules exist, and examples are simply life-savers answering the appeals of rules making vain attempts to exist.”
—André Breton (1896–1966)