In the previous post about Brouwer’s Fixed Point Theorem, we used two black boxes. In this post we will prove the slight variation of those black boxes. We will start with the simplest lemma first: the reduced homology of balls.

**Lemma 2 (Reduced homology of balls)**

Given a

*Proof.* Observe that

In the reduced homology, therefore

**Corollary 1 (Reduced Betti numbers of balls)**

The

Now, we are ready to prove the main theme of this post.

**Lemma 1 (Reduced Betti numbers of spheres)**

Given a

*Proof.* We use “divide-and-conquer” approach to apply Mayer-Vietoris Theorem. We cut the sphere along the equator and note that the upper and lower portion of the sphere is just a disk, and the intersection between those two parts is a circle (sphere one dimension down), as shown in the figure below.

By Mayer-Vietoris Theorem, we have a long exact sequence in the form of:

By Corollary 1,

## References

- Hatcher, Allen. “Algebraic topology.” (2001).