Adjoining a 2-cell

Section 2.17 Adjoining a 2-cell

Definition 2.17.1. 2-cell.

A 2-cell is a space \(B\) that is homeomorphic to the closed disc \(B^2\text{.}\) Given a 2-cell \(B\) with homeomorphism \(\phi\colon S^1\rightarrow B\text{,}\) we denote by \(\operatorname{Bd} B\) the image \(\phi(S^1)\) of the boundary of \(B^2\text{;}\) and we denote by \(\operatorname{Int} B\) the set \(B-\operatorname{Bd} B\text{,}\) which is the image of the open disc \(B^2-S^1\) under \(\phi\text{.}\)

Theorem 2.17.2. Adjoining a 2-cell.

Let \(A\) be a closed path-connected subspace of the Hausdorff space \(X\text{,}\) and suppose there is a 2-cell \(B\) and map \(h\colon B\rightarrow X\) such that (i) \(h(\operatorname{Bd} B)\subseteq A\) and (ii) \(h\vert_{\operatorname{Int} B}\) is a bijection onto \(X-A\text{.}\) Fix any point \(p\in \operatorname{Bd} B\text{,}\) let \(a=h(p)\text{,}\) and define \(k\colon (\operatorname{Bd} B,p)\rightarrow (A,a)\) to be the restriction of \(h\) to \(\operatorname{Bd} B\text{.}\)

The map \(i_*\colon \pi_1(A,a)\rightarrow \pi_1(X,a)\) induced by the inclusion \(A\subseteq X\) is surjective and its kernel \(N\) is the least normal subgroup of \(\pi_1(A,a)\) containing \(k_*(\pi_1(\operatorname{Bd} B, p)\text{.}\) Equivalently, if \(f\in P(\operatorname{Bd} B; p,p)\) is a loop such that \([f]\) generated \(\pi_1(\operatorname{Bd} B, p)\text{,}\) then \(\ker i_*\) is the least normal subgroup of \(\pi_1(A,a)\) containing \(k_*([f])\text{.}\)

Definition 2.17.3. Adjoining a 2-cell.

Let \(A\) be a closed path-connected subspace of the Hausdorff space \(X\text{.}\) We say \(X\) is obtained from \(A\) by adjoining a 2-cell if there is a 2-cell \(B\) and map \(h\colon B\rightarrow X\) such that \(h(\operatorname{Bd} B)\subseteq A \) and \(h\vert_{\operatorname{Int} B}\) is a bijection onto \(X-A\text{.}\)

Example 2.17.4. Torus revisited (again).

Fix any \(R > 1 \)The map \(q\colon I\times I\rightarrow T\subseteq \R^3\) defined as

\begin{equation*} h(s,t)=\left((R+\cos 2\pi t)\cos 2\pi s, (R+\cos 2\pi t)\sin 2\pi s,\sin \2\pi t\right) \end{equation*}

provides a realization of the torus as a closed subspace of \(\R^3\text{.}\) The space \(B=I\times I\) is a \(2-cell\text{,}\) with \(\operatorname{Bd} B\) equal to the square perimeter of \(I\times I\text{.}\) Let \(A=h(\operatorname{Bd} B)\text{.}\)

Show that \(T\) is obtained from \(A\) by adjoining a \(2\)-cell via \(h\text{.}\)
Use the adjoining a 2-cell theorem to compute \(\pi_1(T,P)\) for any point \(P\in T\text{.}\)

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