Surfaces

1 (Fall ’05) #topology/qual/work

State the classification theorem for surfaces (compact, without boundary, but not necessarily orientable). For each surface in the classification, indicate the structure of the first homology group and the value of the Euler characteristic.

Also, explain briefly how the 2-holed torus and the connected sum \({\mathbb{RP}}^2 # {\mathbb{RP}}^2\) fit into the classification.

2 (Spring ’16) #topology/qual/work

Give a list without repetitions of all compact surfaces (orientable or non-orientable and with or without boundary) that have Euler characteristic negative one.

Explain why there are no repetitions on your list.

3 (Spring ’07) #topology/qual/work

Describe the topological classification of all compact connected surfaces \(M\) without boundary having Euler characteristic \(\chi(M )\geq -2\).

No proof is required.

4 (Spring ’09) #topology/qual/work

How many surfaces are there, up to homeomorphism, which are:

  • Connected,
  • Compact,
  • Possibly with boundary,
  • Possibly nonorientable, and
  • With Euler characteristic -3?

Describe one representative from each class.

5 (Fall ’13) #topology/qual/work

Prove that the Euler characteristic of a compact surface with boundary which has \(k\) boundary components is less than or equal to \(2 - k\).

6 (Spring ’13) #topology/qual/work

What surface is represented by the \(6{\hbox{-}}\)gon with edges identified according to the symbol \(xyzxy^{-1}z^{-1}\) ?

7 (Spring ’15) #topology/qual/work

Let \(X\) be the topological space obtained as the quotient space of a regular \(2n{\hbox{-}}\)gon (\(n \geq 2\)) in \({\mathbb{R}}^2\) by identifying opposite edges via translations in the plane.

First show that X is a compact, orientable surface without boundary, and then identify its genus as a function of \(n\).

8 (Fall ’10) #topology/qual/work

a

Show that any compact connected surface with nonempty boundary is homotopy equivalent to a wedge of circles

Hint: you may assume that any compact connected surface without boundary is given by identifying edges of a polygon in pairs.

b

For each surface appearing in the classification of compact surfaces with nonempty boundary, say how many circles are needed in the wedge from part (a).

Hint: you should be able to do this even if you have not done part (a).

9 (Fall ’04) #topology/qual/work

Let \(M_g^2\) be the compact oriented surface of genus \(g\).

Show that there exists a continuous map \(f : M_g^2 \to S^2\) which is not homotopic to a constant map.

10 (Spring ’11) #topology/qual/work

Show that \({\mathbb{RP}}^2 \lor S^1\) is not homotopy equivalent to a compact surface (possibly with boundary).

11 (Fall ’14) #topology/qual/work

Identify (with proof, but of course you can appeal to the classification of surfaces) all of the compact surfaces without boundary that have a cell decomposition having exactly one 0-cell and exactly two 1-cells (with no restriction on the number of cells of dimension larger than 1).

12 (Fall ’11) #topology/qual/work

For any natural number \(g\) let \(\Sigma_g\) denote the (compact, orientable) surface of genus \(g\).

Determine, with proof, all valued of \(g\) with the property that there exists a covering space \(\pi : \Sigma_5 \to \Sigma_g\) .

Hint: How does the Euler characteristic behave for covering spaces?

13 (Spring ’14) #topology/qual/work

Find all surfaces, orientable and non-orientable, which can be covered by a closed surface (i.e. compact with empty boundary) of genus 2. Prove that your answer is correct.

14 (Spring ’18) #topology/qual/work

a

Write down (without proof) a presentation for \(\pi_1 (\Sigma_2 , p)\) where \(\Sigma_2\) is a closed, connected, orientable genus 2 surface and \(p\) is any point on \(\Sigma_2\) .

b

Show that \(\pi_1 (\Sigma_2 , p)\) is not abelian by showing that it surjects onto a free group of rank 2.

c

Show that there is no covering space map from \(\Sigma_2\) to \(S^1 \times S^1\) . You may use the fact that \(\pi_1 (S^1 \times S^1 ) \cong {\mathbf{Z}}^2\) together with the result in part (b) above.

15 (Fall ’16) #topology/qual/work

Give an example, with explanation, of a closed curve in a surfaces which is not nullhomotopic but is nullhomologous.

16 (Fall ’17) #topology/qual/work

Let \(M\) be a compact orientable surface of genus \(2\) without boundary.

Give an example of a pair of loops \begin{align*}\gamma_0 , \gamma_1 : S^1 \to M\end{align*} with \(\gamma_0 (1) = \gamma_1 (1)\) such that there is a continuous map \(\Gamma: [0, 1] \times S^1 \to M\) such that \begin{align*} \Gamma(0, t) = \gamma_0 (t), \quad \Gamma(1, t) = \gamma_1 (t) {\quad \operatorname{for all} \quad} t \in S^1 ,\end{align*} but such that there is no such map \(\Gamma\) with the additional property that \(\Gamma_s (1) = \gamma_0 (1)\) for all \(s \in [0, 1]\).

(You are not required to prove that your example satisfies the stated property.)

17 (Fall ’18) #topology/qual/work

Let \(C\) be cylinder. Let \(I\) and \(J\) be disjoint closed intervals contained in \(\partial C\).

What is the Euler characteristic of the surface \(S\) obtained by identifying \(I\) and \(J\)?

Can all surface with nonempty boundary and with this Euler characteristic be obtained from this construction?

18 (Spring ’19) #topology/qual/work

Let \(\Sigma\) be a compact connected surface and let \(p_1, \cdots , p_k \in \Sigma\).

Prove that \(H_2 \qty{\Sigma \setminus \cup_{i=1}^k {p_i} } = 0\).

Spring 2021 #topology/qual/work

Let \(\Sigma_{g, n}\) denote the compact oriented surface of genus \(g\) with \(n\) boundary components.

  • Show that \(\Sigma_{0, 3}\) and \(\Sigma_{1, 1}\) are both homotopy equivalent to \(S_1 \vee S^1\).

  • Give a complete classification of pairs \((g, n)\) and \((g', n')\) for which \(\Sigma_{g, n}\) is homotopy equivalent to \(\Sigma_{g', n'}\).

#topology/qual/work