Topology - January 2026

Problem 1

The fundamental group of a (based) space $X$ can also be described as the (based) homotopy classes of maps from the circle $S^1$ into $X$. Analogously, one can define groups ${\pi }_{\mathrm{<mi>n</mi>}}\left(X\right)$ as the group of (based) homotopy classes of maps from $S^{\mathrm{<mi>n</mi>}}$ into $X$.

1. Show that every map $f:S^2\to S^1$ is nullhomotopic. This implies that the group .
2. Prove that not every map $g:S^2\to S^2$ is nullhomotopic. This implies that the group is nonzero.

Problem 2

1. Explain how to construct a CW-complex $X$ with homology groups as follows:
2. For the space $X$ you constructed in part (a), compute the cohomology groups $H^i(X;\mathbb{Z}/3\mathbb{Z})$.
3. For a finitely generated abelian group $G$, prove that $\mathrm{Ext}(\mathbb{Z}/n\mathbb{Z},G)\cong G/nG$.

Problem 3

1. Describe the universal cover of ${\mathrm{ℝP}}^2$.
2. Is there a covering space $W$ of ${\mathrm{ℝP}}^2$ such that ?
3. Let $X$ be a path-connected, locally path-connected, and semi-locally simply connected space. Prove that the universal cover of $X$ is unique up to homeomorphism.

Problem 4

1. Give an example of two spaces that are homotopy equivalent but not homeomorphic. Justify your answer.
2. Give an example of two spaces $X$ and $Y$ such that $H_{\mathrm{<mi>i</mi>}}\left(X\right)\cong H_{\mathrm{<mi>i</mi>}}\left(Y\right)$ for all $i$, but $X$ and $Y$ are not homotopy equivalent. Justify your answer.

Problem 5

Let $X$ be the space obtained from the torus $T^2=S^1\times S^1$ and the unit disk $D^2$ by identifying the circle $S^1\times \mathrm{pt}$ with $\partial D^2$ by a homeomorphism (see picture).

1. Calculate .
2. Calculate the homology groups of $X$.