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In this post (Homotopy Equivalence of Punctured Tori), the author of the first answer states that a surface minus finite number of points is homotopy equivalent to a bouquet of circles. However, it doesn't seem very clear to me.

Is there any source I can look at for the proof or explanation?

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    $\begingroup$ A one dim cw complex is a bouquet of circles. A surface has a cw structure with one twocell. Removing one point in the interior of the twocell allows one to retract the cw complex to the one skeleton, i.e a bouquet of circles. Removing more points from the interior amounts to adding extra 1 cells: to see this just study what happens on a disc. $\endgroup$
    – Thomas Rot
    Nov 9, 2020 at 9:40

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The case $g=0$ being straightforward, let me focus to the case $g \geq 1$.

Let $X=\Sigma_g -\{p_1, \ldots p_k\}$ be the topological space obtained by removing $k$ distinct points from a closed surface of genus $g \geq 1$. By Seifert-Van Kampen theorem, a presentation for the fundamental group of $X$ is $$\pi_1(X)=\langle a_1, \, b_1, \ldots a_g, \, b_g, \, c_1, \ldots, c_k \; | \; c_1c_2 \ldots c_k \Pi [a_i, \, b_i]=1 \rangle \simeq F_{2g+k-1},$$ namely, $\pi_1(X)$ is free on $2g+k-1$ generators.

On the other hand, by classical uniformization theory (look for instance at MO question 254687), the universal cover $\tilde{X}$ is homeomorphic to the hyperbolic plane $\mathbb{H}^2$, that is contractible. This means that $X$ is an aspherical $\mathrm{CW}$-complex, in particular its homotopy type is detected by its fundamental group alone.

Therefore $X$ is homotopically equivalent to a bouquet $\bigvee_{i=1}^{2g+k-1} S^1$ of $2g+k-1$ circles, since this is another aspherical $\mathrm{CW}$-complex with the same fundamental group as $X$.

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    $\begingroup$ Is there a proof/explanation that uses more elementary results? I am self-studying topology, and I see that Seifert-Van Kampen theorem will be near the last chapter in the text I am using. In particular, I am curious about a surface that is not homeomorphic to a sphere, so I thought the classification that says the surface should be connected sum of tori or projective plane would be useful. A visualization of the deformation retract of the connected sum minus some point would be sufficient for me now, but I am having a hard time thinking about it. Do you have any source I can look at? $\endgroup$
    – Quin Morris
    Nov 9, 2020 at 9:56
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    $\begingroup$ If you want an ''elementary" explanation, in my opinion the comment by Thomas Rot points in the right direction. You must consider a model of $\Sigma_g$ given by a polygon with $2g$ sides identified in an appropriate way, and then make punctures on it. Puncturing once allows one to retract the polygon on the boundary, and this gives a bouquet of $2g$ circles. Each further puncture just give a further $1$-cell, just think about the case of a disk. $\endgroup$
    – Francesco Polizzi
    Nov 9, 2020 at 10:03

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