Problem 1.

a. Let $X,Y$$X,Y$X,YX, Y$X,Y$ be topological spaces. Suppose $X$$X$XX$X$ is compact and $Y$$Y$YY$Y$ is Hausdorff. Show that every continuous bijection $f:X\to Y$$f:X\to Y$f:X rarr Yf: X \rightarrow Y$f:X\to Y$ is a homeomorphism.
b. Give an example of two topological spaces $X,Y$$X,Y$X,YX, Y$X,Y$, and a continuous map $f:X\to Y$$f:X\to Y$f:X rarr Yf: X \rightarrow Y$f:X\to Y$ such that $f$$f$ff$f$ is a bijection but not a homeomorphism.

Problem 2. Let $\tau$$\tau$tau\tau$\tau$ be the collection of subsets of ${\mathbb{R}}^2$${\mathbb{R}}^2$R^(2)\mathbb{R}^{2}${\mathbb{R}}^2$ defined as follows: $U\in \tau$$U\in \tau$U in tauU \in \tau$U\in \tau$ if and only if $U=\mathrm{\varnothing }$$U=\mathrm{\varnothing }$U=O/U=\emptyset$U=\mathrm{\varnothing }$ or ${\mathbb{R}}^2\mathrm{\setminus }U$${\mathbb{R}}^2\mathrm{\setminus }U$R^(2)\\U\mathbb{R}^{2} \backslash U${\mathbb{R}}^2\mathrm{\setminus }U$ consists of a (possibly empty) finite union of points and straight lines.
a. Show that $\tau$$\tau$tau\tau$\tau$ is a topology on ${\mathbb{R}}^2$${\mathbb{R}}^2$R^(2)\mathbb{R}^{2}${\mathbb{R}}^2$.
b. Determine whether $\tau$$\tau$tau\tau$\tau$ is Hausdorff.
c. Show that $\tau$$\tau$tau\tau$\tau$ is coarser than the Euclidean topology on ${\mathbb{R}}^2$${\mathbb{R}}^2$R^(2)\mathbb{R}^{2}${\mathbb{R}}^2$.
d. Determine whether $\mathbb{R}\times \{0\}$$\mathbb{R}\times \{0\}$Rxx{0}\mathbb{R} \times\{0\}$\mathbb{R}\times \{0\}$ is a compact subspace of $({\mathbb{R}}^2,\tau )$$\left({\mathbb{R}}^2,\tau \right)$(R^(2),tau)\left(\mathbb{R}^{2}, \tau\right)$({\mathbb{R}}^2,\tau )$.

Problem 3. Let $X$$X$XX$X$ be the set of points in ${\mathbb{R}}^2$${\mathbb{R}}^2$R^(2)\mathbb{R}^{2}${\mathbb{R}}^2$ with at least one irrational coordinate, that is, $X={\mathbb{R}}^2\mathrm{\setminus }{\mathbb{Q}}^2$$X={\mathbb{R}}^2\mathrm{\setminus }{\mathbb{Q}}^2$X=R^(2)\\Q^(2)X=\mathbb{R}^{2} \backslash \mathbb{Q}^{2}$X={\mathbb{R}}^2\mathrm{\setminus }{\mathbb{Q}}^2$. Equip $X$$X$XX$X$ with the subspace topology induced by the Euclidean topology on ${\mathbb{R}}^2$${\mathbb{R}}^2$R^(2)\mathbb{R}^{2}${\mathbb{R}}^2$.
a. Show that $X$$X$XX$X$ is path-connected.
b. Show that the fundamental group of $X$$X$XX$X$ is uncountable.

Problem 4. Let $\mathbb{R}{\mathbb{P}}^n$$\mathbb{R}{\mathbb{P}}^n$RP^(n)\mathbb{R} \mathbb{P}^{n}$\mathbb{R}{\mathbb{P}}^n$ denote the $n$$n$nn$n$-dimensional real projective space, and let ${\mathbb{T}}^m=\underset{m\text{-times}}{\underbrace{{\mathbb{S}}^1\times \cdots \times {\mathbb{S}}^1}}$${\mathbb{T}}^m=\underset{m\text{-times }}{\underbrace{{\mathbb{S}}^1\times \cdots \times {\mathbb{S}}^1}}$T^(m)=ubrace(S^(1)xx cdots xxS^(1)ubrace)_(m"-times ")\mathbb{T}^{m}=\underbrace{\mathbb{S}^{1} \times \cdots \times \mathbb{S}^{1}}_{m \text {-times }}${\mathbb{T}}^m=\underset{m\text{-times }}{\underbrace{{\mathbb{S}}^1\times \cdots \times {\mathbb{S}}^1}}$ denote the $m$$m$mm$m$-dimensional torus.
a. Let $n\ge 2$$n\ge 2$n >= 2n \geq 2$n\ge 2$. Show that every continuous map $f:{\mathbb{R}\mathbb{P}}^n\to {\mathbb{T}}^m$$f:{\mathbb{R}\mathbb{P}}^n\to {\mathbb{T}}^m$f:RP^(n)rarrT^(m)f: \mathbb{R P}^{n} \rightarrow \mathbb{T}^{m}$f:{\mathbb{R}\mathbb{P}}^n\to {\mathbb{T}}^m$ is null-homotopic.
b. Is every continuous map $f:{\mathbb{R}}^1\to {\mathbb{T}}^m$$f:{\mathbb{R}}^1\to {\mathbb{T}}^m$f:R^(1)rarrT^(m)f: \mathbb{R}^{1} \rightarrow \mathbb{T}^{m}$f:{\mathbb{R}}^1\to {\mathbb{T}}^m$ null-homotopic?

Problem 5. Let $\mathcal{C}$$\mathcal{C}$C\mathcal{C}$\mathcal{C}$ and $\ell$$\ell$ℓ\ell$\ell$ be respectively a circle and a straight line in ${\mathbb{R}}^3$${\mathbb{R}}^3$R^(3)\mathbb{R}^{3}${\mathbb{R}}^3$ with $\mathcal{C}\cap \ell =\mathrm{\varnothing }$$\mathcal{C}\cap \ell =\mathrm{\varnothing }$Cnnℓ=O/\mathcal{C} \cap \ell=\emptyset$\mathcal{C}\cap \ell =\mathrm{\varnothing }$. Let $S$$S$SS$S$ be the union of $\mathcal{C}$$\mathcal{C}$C\mathcal{C}$\mathcal{C}$ and $\ell$$\ell$ℓ\ell$\ell$, that is, $S=\mathcal{C}\cup \ell$$S=\mathcal{C}\cup \ell$S=CuuℓS=\mathcal{C} \cup \ell$S=\mathcal{C}\cup \ell$. Compute the fundamental group of ${\mathbb{R}}^3\mathrm{\setminus }S$${\mathbb{R}}^3\mathrm{\setminus }S$R^(3)\\S\mathbb{R}^{3} \backslash S${\mathbb{R}}^3\mathrm{\setminus }S$ in the following two cases:
a. There is a plane $\mathrm{\Pi }$$\mathrm{\Pi }$Pi\Pi$\mathrm{\Pi }$ in ${\mathbb{R}}^3$${\mathbb{R}}^3$R^(3)\mathbb{R}^{3}${\mathbb{R}}^3$ such that $\ell$$\ell$ℓ\ell$\ell$ and $\mathcal{C}$$\mathcal{C}$C\mathcal{C}$\mathcal{C}$ lie in different components of ${\mathbb{R}}^3\mathrm{\setminus }\mathrm{\Pi }$${\mathbb{R}}^3\mathrm{\setminus }\mathrm{\Pi }$R^(3)\\Pi\mathbb{R}^{3} \backslash \Pi${\mathbb{R}}^3\mathrm{\setminus }\mathrm{\Pi }$.
b. There is no plane $\mathrm{\Pi }$$\mathrm{\Pi }$Pi\Pi$\mathrm{\Pi }$ in ${\mathbb{R}}^3$${\mathbb{R}}^3$R^(3)\mathbb{R}^{3}${\mathbb{R}}^3$ such that $\ell$$\ell$ℓ\ell$\ell$ and $\mathcal{C}$$\mathcal{C}$C\mathcal{C}$\mathcal{C}$ lie in different components of ${\mathbb{R}}^3\mathrm{\setminus }\mathrm{\Pi }$${\mathbb{R}}^3\mathrm{\setminus }\mathrm{\Pi }$R^(3)\\Pi\mathbb{R}^{3} \backslash \Pi${\mathbb{R}}^3\mathrm{\setminus }\mathrm{\Pi }$.

Problem 6. Let ${\mathbb{T}}^2={\mathbb{S}}^1\times {\mathbb{S}}^1\subset {\mathbb{R}}^4$${\mathbb{T}}^2={\mathbb{S}}^1\times {\mathbb{S}}^1\subset {\mathbb{R}}^4$T^(2)=S^(1)xxS^(1)subR^(4)\mathbb{T}^{2}=\mathbb{S}^{1} \times \mathbb{S}^{1} \subset \mathbb{R}^{4}${\mathbb{T}}^2={\mathbb{S}}^1\times {\mathbb{S}}^1\subset {\mathbb{R}}^4$ denote the 2-torus, defined as the set of points $w,x,y,z$$w,x,y,z$w,x,y,zw, x, y, z$w,x,y,z$ such that $w^2+x^2=y^2+z^2=1$$w^2+x^2=y^2+z^2=1$w^(2)+x^(2)=y^(2)+z^(2)=1w^{2}+x^{2}=y^{2}+z^{2}=1$w^2+x^2=y^2+z^2=1$, with the product orientation determined by the standard orientation on ${\mathbb{S}}^1$${\mathbb{S}}^1$S^(1)\mathbb{S}^{1}${\mathbb{S}}^1$. Compute ${\int }_{{\mathbb{T}}^2}\sigma$${\int }_{{\mathbb{T}}^2} \sigma$int_(T^(2))sigma\int_{\mathbb{T}^{2}} \sigma${\int }_{{\mathbb{T}}^2}\sigma$, where $\sigma$$\sigma$sigma\sigma$\sigma$ is the following 2-form on ${\mathbb{R}}^4$${\mathbb{R}}^4$R^(4)\mathbb{R}^{4}${\mathbb{R}}^4$ :

a. Given a smooth map $F:M\to N$$F:M\to N$F:M rarr NF: M \rightarrow N$F:M\to N$ between two smooth manifolds $M$$M$MM$M$ and $N$$N$NN$N$ define the notions of a critical point and a critical value of $F$$F$FF$F$.
b. Define $\mathcal{Z}:=\{(x,a,b,c)\in {\mathbb{R}}^3:a{x}^2+bx+c=0,a\ne 0\}$$\mathcal{Z}:=\left\{(x,a,b,c)\in {\mathbb{R}}^3:a{x}^2+bx+c=0,a\ne 0\right\}$Z:={(x,a,b,c)inR^(3):ax^(2)+bx+c=0,a!=0}\mathcal{Z}:=\left\{(x, a, b, c) \in \mathbb{R}^{3}: a x^{2}+b x+c=0, a \neq 0\right\}$\mathcal{Z}:=\{(x,a,b,c)\in {\mathbb{R}}^3:a{x}^2+bx+c=0,a\ne 0\}$.
i) Prove that $\mathcal{Z}$$\mathcal{Z}$Z\mathcal{Z}$\mathcal{Z}$ is a smooth submanifold of ${\mathbb{R}}^3$${\mathbb{R}}^3$R^(3)\mathbb{R}^{3}${\mathbb{R}}^3$;
ii) Define $\pi :\mathcal{Z}\to {\mathbb{R}}^3$$\pi :\mathcal{Z}\to {\mathbb{R}}^3$pi:ZrarrR^(3)\pi: \mathcal{Z} \rightarrow \mathbb{R}^{3}$\pi :\mathcal{Z}\to {\mathbb{R}}^3$ by $\pi (x,a,b,c)=(a,b,c)$$\pi (x,a,b,c)=(a,b,c)$pi(x,a,b,c)=(a,b,c)\pi(x, a, b, c)=(a, b, c)$\pi (x,a,b,c)=(a,b,c)$ for every $(x,a,b,c)\in \mathcal{Z}$$(x,a,b,c)\in \mathcal{Z}$(x,a,b,c)inZ(x, a, b, c) \in \mathcal{Z}$(x,a,b,c)\in \mathcal{Z}$. Prove that $(a,b,c)$$(a,b,c)$(a,b,c)(a, b, c)$(a,b,c)$ is a critical value of $\pi$$\pi$pi\pi$\pi$ if and only of $b^2-4ac=0$$b^2-4ac=0$b^(2)-4ac=0b^{2}-4 a c=0$b^2-4ac=0$.

Problem 8. Consider the set of all ordered triples of vectors in ${\mathbb{R}}^4$${\mathbb{R}}^4$R^(4)\mathbb{R}^{4}${\mathbb{R}}^4$ such that they form an orthonormal basis of their linear span. Endow this set with the natural structure of an embedded submanifold in ${\mathbb{R}}^N$${\mathbb{R}}^N$R^(N)\mathbb{R}^{N}${\mathbb{R}}^N$ for certain $N$$N$NN$N$. What is the dimension of this submanifold?

Problem 9. Given $g_1,g_2\in C^{\mathrm{\infty }}\left({\mathbb{R}}^2\right)$$g_1,g_2\in C^{\mathrm{\infty }}\left({\mathbb{R}}^2\right)$g_(1),g_(2)inC^(oo)(R^(2))g_{1}, g_{2} \in C^{\infty}\left(\mathbb{R}^{2}\right)$g_1,g_2\in C^{\mathrm{\infty }}\left({\mathbb{R}}^2\right)$ and two smooth vector fields $X_1$$X_1$X_(1)X_{1}$X_1$ and $X_2$$X_2$X_(2)X_{2}$X_2$ in ${\mathbb{R}}^2$${\mathbb{R}}^2$R^(2)\mathbb{R}^{2}${\mathbb{R}}^2$ such that $X_1$$X_1$X_(1)X_{1}$X_1$ and $X_2$$X_2$X_(2)X_{2}$X_2$ are linearly independent at every point (i.e., form a frame) and $[X_1,X_2]={\alpha }_1{X}_1+{\alpha }_2{X}_2$$\left[X_1,X_2\right]={\alpha }_1{X}_1+{\alpha }_2{X}_2$[X_(1),X_(2)]=alpha_(1)X_(1)+alpha_(2)X_(2)\left[X_{1}, X_{2}\right]=\alpha_{1} X_{1}+\alpha_{2} X_{2}$[X_1,X_2]={\alpha }_1{X}_1+{\alpha }_2{X}_2$ for some smooth functions ${\alpha }_1$${\alpha }_1$alpha_(1)\alpha_{1}${\alpha }_1$ and ${\alpha }_2$${\alpha }_2$alpha_(2)\alpha_{2}${\alpha }_2$ in ${\mathbb{R}}^2$${\mathbb{R}}^2$R^(2)\mathbb{R}^{2}${\mathbb{R}}^2$, prove that the following system of equations with respect to the function $u\in C^{\mathrm{\infty }}\left({\mathbb{R}}^2\right)$$u\in C^{\mathrm{\infty }}\left({\mathbb{R}}^2\right)$u inC^(oo)(R^(2))u \in C^{\infty}\left(\mathbb{R}^{2}\right)$u\in C^{\mathrm{\infty }}\left({\mathbb{R}}^2\right)$

has a solution on ${\mathbb{R}}^2$${\mathbb{R}}^2$R^(2)\mathbb{R}^{2}${\mathbb{R}}^2$ if and only if

Problem 10. Let $D$$D$DD$D$ be a distribution on a manifold M, $X$$X$XX$X$ be a vector field on $M$$M$MM$M$, and $e^{tX}$$e^{tX}$e^(tX)e^{t X}$e^{tX}$ denote the local flow of $X$$X$XX$X$.
a. Prove that if ${\left(e^{tX}\right)}_{\ast }D=D$${\left(e^{tX}\right)}_{\ast }D=D$(e^(tX))_(**)D=D\left(e^{t X}\right)_{*} D=D${\left(e^{tX}\right)}_{\ast }D=D$ for sufficiently small $t$$t$tt$t$ then

i.e. the vector field $[X,Y]$$[X,Y]$[X,Y][X, Y]$[X,Y]$ is tangent to $D$$D$DD$D$ for every vector field $Y$$Y$YY$Y$ tangent to $D$$D$DD$D$.
b. A vector field $X$$X$XX$X$ is called an infinitesimal symmetry of the distribution $D$$D$DD$D$ if the relation (1) holds. Assume that $\dim M=n$$\dim M=n$dim M=n\operatorname{dim} M=n$\dim M=n$, $\mathrm{rank}D=m$$\mathrm{rank}D=m$rank D=m\operatorname{rank} D=m$\mathrm{rank}D=m$. Prove that $X$$X$XX$X$ is an infinitesimal symmetry of $D$$D$DD$D$ if and only if for every tuple of the locally defining 1 - forms ${\omega }_1,\dots {\omega }_{n-m}$${\omega }_1,\dots {\omega }_{n-m}$omega_(1),dotsomega_(n-m)\omega_{1}, \ldots \omega_{n-m}${\omega }_1,\dots {\omega }_{n-m}$ of $D$$D$DD$D$ the following identities hold:

c. Let $M={\mathbb{R}}^3$$M={\mathbb{R}}^3$M=R^(3)M=\mathbb{R}^{3}$M={\mathbb{R}}^3$ with standard coordinates $(x,y)$$(x,y)$(x,y)(x, y)$(x,y)$, and $D=\ker \theta$$D=\ker \theta$D=ker thetaD=\operatorname{ker} \theta$D=\ker \theta$

Prove that for any $f\in C^{\mathrm{\infty }}\left({\mathbb{R}}^3\right)$$f\in C^{\mathrm{\infty }}\left({\mathbb{R}}^3\right)$f inC^(oo)(R^(3))f \in C^{\infty}\left(\mathbb{R}^{3}\right)$f\in C^{\mathrm{\infty }}\left({\mathbb{R}}^3\right)$ there exists the unique infinitesimal symmetry $X_f$$X_f$X_(f)X_{f}$X_f$ of $D$$D$DD$D$ such that $\theta \left(X_f\right)\equiv f$$\theta \left(X_f\right)\equiv f$theta(X_(f))-=f\theta\left(X_{f}\right) \equiv f$\theta \left(X_f\right)\equiv f$. Express $X_f$$X_f$X_(f)X_{f}$X_f$ explicitly in terms of $f$$f$ff$f$.