Examples

Contraction to a point

Contraction to a point

The map $f:{\mathbb{R}}^n\to \{p\}$ for some point $p\in {\mathbb{R}}^n$ is a homotopy equivalence, so it induces an isomorphism on homology. Therefore:

$$H_i({\mathbb{R}}^n)=\{\begin{array}{ll}0& i>0\\ \mathbb{Z}& i=0\end{array}$$

Dunce Hat

Dunce Hat

The Dunce Hat is formed by identifying all sides of a triangle with non-circular identifications:

The Dunce Hat is contractible, but it is not collapsible. The contraction can be performed by embedding the Hat in a 3-ball in ${\mathbb{R}}^4$.

Because it is contractible, the Dunce Hat has the homology of a point.

Homology of a cylinder

Homology of a cylinder

Recall that the cylinder is homotopic to the circle (§06):

A circle is homeomorphic to a triangle.

We have computed the homology of a triangle (§11 of Part I).

Therefore the homology of the cylinder is:

$$H_i(X)=\{\begin{array}{ll}0& i>2\\ \mathbb{Z}& i=1\\ \mathbb{Z}& i=0\end{array}$$