# How do we know that a surface minus finite number of points is homotopy equivalent to a bouquet of circles? [closed]

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?

• 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. Nov 9 '20 at 9:40

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 $$X$$.
• 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. Nov 9 '20 at 10:03