\documentclass[tikz]{standalone}

\usepackage{amsmath, amsthm, amssymb, amsfonts, mathrsfs, mathtools}
\usepackage{tikz}
\usepackage{amsmath}
\usepackage{color}

\usetikzlibrary{patterns}
\usetikzlibrary{scopes}

\begin{document}

    \begin{tikzpicture}
        % Sets in R^m

        \draw[fill = blue!50!white,smooth cycle] plot coordinates{(-2, -4.5) (-2, -3.2) (-0.8, -3.2) (-0.8, -4.5)};
        \draw[pattern color=black, pattern=north west lines] {(-0.67, -3.06) -- +(180:0.8) arc (180:270:0.8) -- cycle};
        \fill[even odd rule, white] {plot coordinates{(-3, -5) (-3, -2) (-0.5, -2) (-0.5, -5)}} {plot [smooth cycle] coordinates{(-2, -4.5) (-2, -3.2) (-0.8, -3.2) (-0.8, -4.5)}};

        %\draw[blue!50!white,fill=blue!50!white] (5.7, -3.06) -- +(-90:0.8) arc (-90:0:0.8);
        \fill[blue!50!white] [smooth cycle] (5.65, -3.05) -- +(-91:0.8) arc (-94:0:0.8);
        \fill[even odd rule, white] [smooth cycle] plot coordinates{(7, -4.5) (7, -3.2) (5.8, -3.2) (5.8, -4.5)} (5.65, -3.05) -- +(-91:0.8) arc (-94:0:0.8);
        \fill[even odd rule, white] [smooth cycle] plot coordinates{(7, -4.5) (7, -3.2) (5.8, -3.2) (5.8, -4.5)} (5.65, -3.05) -- +(-91:0.8) arc (-94:0:0.8);
        \fill[even odd rule, white] [smooth cycle] plot coordinates{(7, -4.5) (7, -3.2) (5.8, -3.2) (5.8, -4.5)} (5.65, -3.05) -- +(-91:0.8) arc (-94:0:0.8);
        \fill[even odd rule, white] [smooth cycle] plot coordinates{(7, -4.5) (7, -3.2) (5.8, -3.2) (5.8, -4.5)} (5.65, -3.05) -- +(-91:0.8) arc (-94:0:0.8);
        \fill[even odd rule, white] [smooth cycle] plot coordinates{(7, -4.5) (7, -3.2) (5.8, -3.2) (5.8, -4.5)} (5.65, -3.05) -- +(-91:0.8) arc (-94:0:0.8);
        \draw[smooth cycle, pattern color=black, pattern=north west lines] plot coordinates{(7, -4.5) (7, -3.2) (5.8, -3.2) (5.8, -4.5)};
        \draw (5.68, -3.857) arc (-91:-1:0.8);

        % Manifold
        \draw[smooth cycle, tension=0.4, fill=white, pattern color=white, opacity=1] plot coordinates{(2,2) (-0.5,0) (3,-2) (5,1)} node at (3,2.3) {$M$};
        %\draw[white, fill=white] (0.95,-1.2) circle (.15cm);
        \draw[white, fill=white] (4.54,-0.12) circle (.15cm);

        % Functions i
        \draw[white,fill=white] (0.06,-0.57) circle (.15cm);
        \path[->] (0.8, 0) edge [bend right] node[left, xshift=-2mm] {$\phi_i$} (-1, -2.9);
        %\path[->] (-0.7, -3.05) edge [bend right] node [right, yshift=-3mm] {$\phi^{-1}_i$} (1.093, -0.11);


        % Functions j
        %\path[->] (5.8, -2.8) edge [bend left] node[midway, xshift=-5mm, yshift=-3mm] {$\phi^{-1}_j$} (3.8, -0.35);
        %\draw[white, fill=white] (4,-1.1) circle (.15cm);
        \path[->] (4.2, 0) edge [bend left] node[right, xshift=2mm] {$\phi_j$} (6.2, -2.8);

        % Help lines
        %\draw[help lines] (-3,-6) grid (8,6);

        % Subsets
        \filldraw[smooth cycle,fill=blue!50!white,draw=black]
            plot coordinates {(1,0) (1.5, 1.2) (2.5,1.3) (2.6, 0.4)}
            node [label={[label distance=-0.3cm, xshift=-2cm, fill=white]:$U_i$}] {};
        \draw[smooth cycle, pattern color=black, pattern=north west lines]
            plot coordinates {(4, 0) (3.7, 0.8) (3.0, 1.2) (2.5, 1.2) (2.2, 0.8) (2.3, 0.5) (2.6, 0.3) (3.5, 0.0)}
            node [label={[label distance=-0.8cm, xshift=.75cm, yshift=1cm, fill=white]:$U_j$}] {};

        % First Axis
        %\draw[thick, ->] (-3,-5) -- (0, -5) node [label=above:$\phi_i(U_i)$] {};
        %\draw[thick, ->] (-3,-5) -- (-3, -2) node [label=right:$\R^n$] {};

        % Arrow from i to j
        \draw[->] (0, -3.85) -- node[midway, above]{$\phi_j\circ \phi_i^{-1}\in G$} (4.8, -3.85);

        % Second Axis
        \draw node at (7.3,-3) {$X$};

        \draw node at (-2.3,-3) {$X$};
        %\draw[thick, ->] (5, -5) -- (8, -5) node [label=above:$\phi_j(U_j)$] {};
        %\draw[thick, ->] (5, -5) -- (5, -2) node [label=right:$\R^n$] {};

        % Hole
        \begin{scope}[scale=.8]
        \path[rounded corners=24pt] (-.9+3,-1+0)--(0+3,-1+.6)--(.9+3,-1+0) (-.9+3,-1+0)--(0+3,-1+-.56)--(.9+3,-1+0);
        \draw[rounded corners=28pt] (-1.1+3,-1+.1)--(0+3,-1+-.6)--(1.1+3,-1+.1);
        \draw[rounded corners=24pt] (-.9+3,-1+0)--(0+3,-1+.6)--(.9+3,-1+0);
      \end{scope}

    \end{tikzpicture}


\end{document}