Long Exact Sequence of a Good Pair

Long Exact Sequence of a Good Pair

For a good pair , there is a long exact sequence in homology: The maps are:

• : induced by the inclusion map .
• : induced by the quotient map .
• : the boundary map.

Optionally, we can write the sequence in reduced homology as: But note that is always reduced.

Proof This can be concluded from

Applications

Corollary

The reduced homology of the n-sphere is given by:

Proof We will prove by induction on . Base Case (n=0): consists of two points, . . The reduced homology is and for . The formula holds. Inductive Step: Assume the formula holds for . Consider the good pair . The quotient space is homeomorphic to . The long exact sequence in reduced homology is: Since the disk is contractible, its reduced homology groups are all zero: for all . The sequence simplifies to: Exactness implies that the connecting homomorphism is an isomorphism for all . By the inductive hypothesis, is for (i.e., ) and otherwise. Therefore, the same is true for , completing the induction.

The long exact sequence can be used to prove Brouwer’s fixed point theorem for higher dimensions:

Brouwer's Fixed Point Theorem

Any continuous map has a fixed point.

Proof Assume there exists a map with no fixed points. We can then define a retraction by sending a point to the intersection of the ray from through with the boundary sphere (just like what we did here). Let be the inclusion map. The composition is the identity map. Note that is a good pair, so is an exact sequence. However, , and , so it is impossible to compose to the identity, yielding a contradiction.