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Find a degree one map ${𝑆}^1\times {𝑆}^2\to ℝ{\mathbb{P}}^3$ or prove that no such map exists.

Consider $𝑋=ℂ{\mathbb{P}}^1\times ℂ{\mathbb{P}}^1$ and $𝑌=ℂ{\mathbb{P}}^2\vee ℂ{\mathbb{P}}^1$.

(a) for any abelian coefficient group $𝐺$ compute the homology groups ${𝐻}_{\ast }(𝑋;𝐺)$ and ${𝐻}_{\ast }(𝑌;𝐺)$.

(b) show that $𝑋$ and $𝑌$ are not homotopy equivalent.

Let $𝜔$ be a volume form on ${𝑆}^2$. For each map $𝑓:{𝑆}^3\to {𝑆}^2$, the form ${𝑓}^{\ast }𝜔$ is a closed 2-form, so we can write ${𝑓}^{\ast }𝜔=d𝛼$ for some 1-form $𝛼$. The number

is called the Hopf invariant of $𝑓$.

(a) Prove that $𝐻(𝑓)$ does not depend on the choice of $𝛼$.

(b) Prove that homotopic maps have the same Hopf invariant.

Show that the tangent space of $ℝ{\mathbb{P}}^2$ does not admit a flat connection.

Let $𝑓:𝑀\to ℝ$ be a Morse function on a 2-dimensional closed oriented surface, and let $𝜑:{𝑆}^1\to 𝑀$ be an embedding of a circle. Show that $𝜑$ can be homotoped to an embedding $𝜓:{𝑆}^1\to 𝑀$ such that $𝑓\circ 𝜓$ is Morse.

Let $𝑀$ be a closed oriented 3-manifold that smoothly embeds in ${ℝ}^4$, and such that ${𝐻}^1(𝑀;\mathbb{Z})=0$. Show that ${𝐻}_1(𝑀;\mathbb{Z})$ is of the form for some abelian group $𝐺$.

If ${ℝ}^{𝑘+𝑟}\to {ℝ}^{𝑛}$ is the projection to the last $𝑛$-coordinates, explicitly describe the integration map ${\mathrm{\Omega }}_{𝑐}^{\ast }({ℝ}^{𝑛+𝑟})\to {\mathrm{\Omega }}_{𝑐}^{\ast -𝑟}({ℝ}^{𝑛})$, and show that it is a cochain map of degree $-𝑟$.

Consider a (regular) octagon whose edges are labeled $\{{𝑒}_0,\dots ,{𝑒}_7\}$ consecutively around the boundary. Identify the edges ${𝑒}_0$ and ${𝑒}_4$, as well as ${𝑒}_2$ and ${𝑒}_6$ by homeomorphisms.

1. Show that the quotient space is a topological manifold with boundary.

2. Identify the choices which determine the homeomorphism type of the resulting surface.

3. Describe all such possible homeomorphism types (your answer should take the form: complement of $𝑘$-discs from either (i) a closed surface of genus $𝑔$ in the orientable case, or (ii) a connected sum of $ℎ$ copies of $ℝ{\mathbb{P}}^2$ in the unorientable case).

Let $𝐿$ be an integrable distribution on a smooth manifold $𝑀$, and let $𝛼$ be a nowhere vanishing 1-form. Show that, if $𝐿$ is transverse to the kernel of $𝛼$, and the restriction of $𝛼$ to the leaves of the foliation defined by $𝐿$ is closed, then the intersection $𝐿\cap ker(𝛼)$ is integrable.

Let ${𝑋}_{{𝑚}_1,{𝑚}_2}^{𝑛}$ be the space obtained from a sphere ${𝑆}^{𝑛}$ by attaching two $𝑛+1$ cells: one by a degree ${𝑚}_1$ map and the other by a degree ${𝑚}_2$ map. Assuming that ${𝑚}_1$ and ${𝑚}_2$ do not vanish, compute the reduced homology and cohomology groups of ${𝑋}_{{𝑚}_1,{𝑚}_2}^{𝑛}$ with integral coefficients, as well as the maps induced by the inclusion of ${𝑆}^{𝑛}$.