Math Problems Vault

Problem 2.5.1

Given two compact connected surfaces $M, N$ with boundary, choose an embedding $f$ of $\left\{ \pm 1\right\} \times D^{1}$ which sends $\left\{ -1\right\} \times D^{1}$ into an arc in $\partial M$ and sends $\left\{ 1\right\} \times D^{1}$ into an arc in $\partial N$ . The boundary connected sum of $M$ and $N$ , written $M \mathbin {\overline{\cup }} N$ , is the surface $M \cup_{f} D^{1} \times D^{1} \cup_{f} N$ formed by attaching a $1$ -handle $D^1 \times D^1$ joining $M$ and $N$ along these arcs.

If we choose different embeddings $i_{M}, i_{N}$ (in the analogous construction for the connected sum $M \# N$ , where a disk is removed from each of $M, N$ and a cylinder $D^{1} \times S^{1}$ is glued in along the two resulting boundary circles) giving $i_{M}', i_{N}'$ , then there are orientation-preserving homeomorphisms $h_{M}: M \to M$ , $h_{N}: N \to N$ with $h_{M} i_{M} = i_{M}'$ , $h_{N} i_{N} = i_{N}'$ , induced by isotopies connecting $i_{M}, i_{M}'$ and $i_{N}, i_{N}'$ . These induce a homeomorphism between the two ways of forming the connected sum, by sending $M \setminus \left\{ i_{M}(0)\right\}$ to $M \setminus \left\{ i_{M}'(0)\right\}$ via $h_{M}$ and $N \setminus \left\{ i_{N}(0)\right\}$ to $N \setminus \left\{ i_{N}'(0)\right\}$ via $h_{N}$ .

Verify that this description of the homeomorphism in pieces does fit together to give a homeomorphism between the two ways of forming the connected sum with $i_{M}, i_{N}$ and $i_{M}', i_{N}'$ ; that is, check that it is consistent with the identifications being made to form the connected sums.

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Problem 2.5.2

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 2.5.3

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

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Problem 2.5.4

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 2.5.5

Show that $M \# N \simeq N \# M$ , where $\#$ denotes connected sum of surfaces (possibly without boundary).

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Problem 2.5.6

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

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Problem 2.5.7

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

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

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

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

(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 $A^{(k)}_{(p)}$ .)