Math Problems Vault

Problem 2.2.5 from Lawson T. "Topology: A Geometric Approach" [Law06]More problems in Section 2.2 "Handle decompositions and more basic surfaces"

Show by induction that if $D_{1}, \ldots , D_{k}$ are $k$ disjoint disks in the interior of a connected surface $M$ , and $D_{1}', \ldots , D_{k}'$ is another collection of $k$ disjoint disks in the interior of $M$ , then there is a homeomorphism $h: M \to M$ which is the identity on $\partial M$ sending $D_{i}$ to $D_{i}'$ .

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Remark.

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This is the ``disk lemma'': for disks $D_{1}, D_{2}$ in the interior of a connected surface (with boundary) $M$ , there is a homeomorphism $h: M \to M$ , fixed on $\partial M$ , with $h(D_{1}) = D_{2}$ .

Problem 2.5.2 from Lawson T. "Topology: A Geometric Approach" [Law06]More problems in Section 2.5 "Connected sums"

Construct a homeomorphism from $[-1,1] \times [-1,1]$ onto $[-1,1] \times [-1,0]$ which fixes pointwise $\left\{ -1\right\} \times [-1,0] \cup [-1,1] \times \left\{ -1\right\} \cup \left\{ 1\right\} \times [-1,0]$ .

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Problem 2.5.4 from Lawson T. "Topology: A Geometric Approach" [Law06]More problems in Section 2.5 "Connected sums"

Show that $(M \mathbin {\overline{\cup }} N) \mathbin {\overline{\cup }} Q \simeq M \mathbin {\overline{\cup }} (N \mathbin {\overline{\cup }} Q)$ , where $\mathbin {\overline{\cup }}$ denotes boundary connected sum of surfaces with boundary (Exercise 2.5.1).

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Problem 2.5.6 from Lawson T. "Topology: A Geometric Approach" [Law06]More problems in Section 2.5 "Connected sums"

Show that $(M \# N) \# Q \simeq M \# (N \# Q)$ .

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Problem 2.5.7 from Lawson T. "Topology: A Geometric Approach" [Law06]More problems in Section 2.5 "Connected sums"

Show that $M \mathbin {\overline{\cup }} D^{2} \simeq M$ , where $\mathbin {\overline{\cup }}$ denotes boundary connected sum (Exercise 2.5.1) and $D^{2}$ is the closed disk.

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Remark.

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Write $M \mathbin {\overline{\cup }} D^{2}$ as $M \cup R_{1} \cup R_{2}$ , where $R_{1}, R_{2}$ are rectangles joined along a segment in their boundaries, and use the idea of Exercise 2.5.2.

Problem 2.5.8 from Lawson T. "Topology: A Geometric Approach" [Law06]More problems in Section 2.5 "Connected sums"

If $N$ is a surface without boundary, and $N_{(p)}$ denotes $N$ with $p$ disjoint disks removed from its interior, show that $N_{(p)} \mathbin {\overline{\cup }} S_{(q)} \simeq N_{(p+q-1)}$ , where $S$ is the $2$ -sphere and $\mathbin {\overline{\cup }}$ denotes boundary connected sum (Exercise 2.5.1).

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Remark.

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Consider first the case $q = 1$ .

Problem 2.5.9 from Lawson T. "Topology: A Geometric Approach" [Law06]More problems in Section 2.5 "Connected sums"

Show that $M \# S^{2} \simeq M$ , where $S^{2}$ is the $2$ -sphere.

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Remark.

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Think about what is removed and what is filled in when we form the connected sum with $S^{2}$ . Alternatively, use Exercise 2.5.7 together with the fact that $M \# N \simeq (M_{(1)} \mathbin {\overline{\cup }} N_{(1)}) \cup h^{2}$ , where $M_{(1)}, N_{(1)}$ denote $M, N$ with one disk removed, $\mathbin {\overline{\cup }}$ is boundary connected sum, and the final $2$ -handle fills in the one remaining boundary circle.

Problem 2.5.10 from Lawson T. "Topology: A Geometric Approach" [Law06]More problems in Section 2.5 "Connected sums"

If $K$ is the Klein bottle, use $K \simeq P \# P$ (where $P$ is the projective plane) to show that $K_{(1)} \simeq P_{(1)} \mathbin {\overline{\cup }} P_{(1)}$ , where the subscript $(1)$ denotes removing one disk and $\mathbin {\overline{\cup }}$ is boundary connected sum (Exercise 2.5.1).

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Problem 2.5.11 from Lawson T. "Topology: A Geometric Approach" [Law06]More problems in Section 2.5 "Connected sums"

(a)(b)

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(a)

Show that $A \# B \# C \simeq (A_{(1)} \mathbin {\overline{\cup }} B_{(1)} \mathbin {\overline{\cup }} C_{(1)}) \cup h^{2}$ , where $\mathbin {\overline{\cup }}$ is boundary connected sum and the subscript $(1)$ denotes removing one disk. Use this to show that the boundary connected sum of $k$ copies of $A_{(1)}$ is homeomorphic to the connected sum of $k$ copies of $A$ with one disk removed; that is, denoting the connected sum of $k$ copies of $A$ by $A^{(k)}$ , show that $A^{(k)}_{(1)} \simeq A_{(1)} \mathbin {\overline{\cup }} \cdots \mathbin {\overline{\cup }} A_{(1)}$ ( $k$ copies).

(b)

Then show that $A^{(k)} \mathbin {\overline{\cup }} S_{(p)} \simeq A_{(1)} \mathbin {\overline{\cup }} \cdots \mathbin {\overline{\cup }} A_{(1)} \mathbin {\overline{\cup }} S_{(p)}$ ( $k$ copies of $A_{(1)}$ ), where $S$ is the $2$ -sphere. (This space is denoted $A^{(k)}_{(p)}$ .)

Problem 2.7.2 from Lawson T. "Topology: A Geometric Approach" [Law06]More problems in Section 2.7 "Euler characteristic and the identification of surfaces"

(a)(b)

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(a)

Show that $\chi (A_{1} \mathbin {\overline{\cup }} \cdots \mathbin {\overline{\cup }} A_{n}) = \chi (A_{1}) + \cdots + \chi (A_{n}) - (n-1)$ , where $\mathbin {\overline{\cup }}$ denotes boundary connected sum.

(b)

Show that $\chi (A_{1} \# \cdots \# A_{n}) = \chi (A_{1}) + \cdots + \chi (A_{n}) - 2(n-1)$ .

Problem 2.9.47 from Lawson T. "Topology: A Geometric Approach" [Law06]More problems in Section 2.9 "Supplementary exercises"

Write the surfaces in Figure 2.71 as connected sums, or boundary connected sums, of other surfaces, none of which is a disk or a sphere. Indicate in the figure what the two pieces are and which of the standard surfaces (with boundary) each is homeomorphic to.

Connected sums.

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Problem 2.9.48 from Lawson T. "Topology: A Geometric Approach" [Law06]More problems in Section 2.9 "Supplementary exercises"

In each of the surfaces with boundary in Figure 2.72, find an arc $A$ joining two points on the boundary so that if we cut the surface along the arc and open it up (equivalently, remove a neighborhood $A \times (-1,1)$ of the arc), we get two surfaces, neither of which is homeomorphic to a disk. Relate this to a boundary connected-sum decomposition.

Separating arcs and boundary sums.

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Problem 2.9.49 from Lawson T. "Topology: A Geometric Approach" [Law06]More problems in Section 2.9 "Supplementary exercises"

In each of the surfaces in Figure 2.73, find an embedded circle $C$ which separates the surface so that if we cut the surface along the circle (equivalently, remove a neighborhood $C \times (-1,1)$ of the circle), we get two surfaces, neither of which is homeomorphic to $S_{(p)}$ (a sphere with some disks removed). Relate this to a connected-sum decomposition.

Separating circles and connected sums.

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Problem 2.9.74 from Lawson T. "Topology: A Geometric Approach" [Law06]More problems in Section 2.9 "Supplementary exercises"

Identify the surface obtained by taking the connected sum of two disks. Identify the surface obtained by taking the boundary connected sum of two disks.

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Connected sums of surfaces

References