querent

Let $𝐺$ be a finite group acting freely on a connected manifold $𝑀$. Construct an associated surjective map ${𝜋}_1(𝑀/𝐺)\to 𝐺$, and show that the composite ${𝜋}_1(𝑀)\to {𝜋}_1(𝑀/𝐺)\to 𝐺$ is trivial.

Construct a pair of topological spaces $𝑋$ and $𝑌$ with isomorphic fundamental groups and homology groups, but whose universal covers have non-isomorphic homology.

Let ${𝑀}_{+}$ be a compact submanifold of ${𝑆}^{𝑛}$, whose complement deformation retracts to a compact submanifold ${𝑀}_{-}$. If $𝑛\ne 1$, show that ${𝑀}_{+}\bigsqcup {𝑀}_{-}$ has exactly two components which do not have codimension 1, and provide, for each dimension $𝑛\ne 1$ and integer $𝑘$, an example where the number of codimension 1 components is $𝑘$.

Give an example of a Morse function on the complex projective space $ℂ{\mathbb{P}}^{𝑛}$, and calculate the Morse indices of its critical points.

Let $SO(𝑛-1)\to {𝑆}^{𝑛-1}$ denote the map which assigns to a matrix its first column. Show that the following two statements are equivalent:

1. There exists a rank-$𝑛$ vector bundle on ${𝑆}^{𝑛}$ whose Euler class is the product of a generator of ${𝐻}^{𝑛}({𝑆}^{𝑛};\mathbb{Z})$ with $𝑑$.

2. There exists a continuous map $𝛼:{𝑆}^{𝑛-1}\to SO(𝑛)$ so that the composite ${𝑆}^{𝑛-1}\to SO(𝑛)\to {𝑆}^{𝑛-1}$ has degree $𝑑$.

Let $𝑓:𝑀\to {ℝ}^{𝑛}$ be an immersion. An ordered double point of $𝑓$ is a pair $(𝑥,𝑦)\in 𝑀\times 𝑀$, with $𝑥\ne 𝑦$, satisfying the property that $𝑓(𝑥)=𝑓(𝑦)$. Show that the set of ordered double points is closed.

Let $𝑋$ be the wedge sum ${𝑆}^1\vee {𝑆}^1$. Give an example of a connected non-regular covering $\widetilde{𝑋}\to 𝑋$, and compute its group of deck transformations.

Show that the spaces ${𝑆}^2\times {𝑆}^4$ and $ℂ{\mathbb{P}}^3$ have isomorphic integral cohomology groups but are not homotopy equivalent.

Let $𝐾$ denote the Klein bottle. Find the minimum number of critical points of a Morse function $𝑓:𝐾\to ℝ$.

Let $𝛾$ be a smooth embedded curve in the plane ${ℝ}^2$. Let $𝐴$ be the set of all values $𝑟>0$ such that the circle of radius $𝑟$ about the origin is tangent to the curve $𝛾$ at some point. Show that the interior of $𝐴$ (as a subset of $ℝ$) is empty.