Fix a positive integer \(n\text{,}\) let \(\alpha=2\pi/n\text{,}\) and let \(f_\alpha\colon S^1\rightarrow S^1\) be the rotation by \(\alpha\) map: i.e., for a point \(P=(\cos \theta, \sin \theta)\in S^1\text{,}\) we have \(f_\alpha(P)=(\cos(\theta+2\pi/n), \sin(\theta+2\pi/n))\text{.}\)
The \(n\)-fold dunce cap is the quotient \(X\) of \(B^2\) obtained by identifying any two points \(P,Q\in S^1\) satisfying \(f_\alpha(P)=f_{\alpha}(Q)\text{.}\)
Lemma2.18.2.Normality of quotients.
Let \(q\colon X\rightarrow Y\) be a closed quotient map. If \(X\) is normal, then \(Y\) is normal.
Proof.
First we show that \(Y\) is a \(T_1\)-space. Since \(q\) is surjective, for all \(y\in Y\) we have \(y=q(x)\) for some \(x\in X\text{.}\) Since \(X\) is \(T_1\text{,}\) the set \(\{x\}\) is closed. Since \(q\) is a closed map, the set \(\{y}=q(\{x\})\) is closed. Thus \(Y\) is \(T_1\text{.}\)
We use the alternate formulation of normality. Consider an inclusion \(A\subseteq U\subseteq Y\text{,}\) where \(A\) is closed and \(U\) is open. Taking preimages under \(q\) yields an inclusion \(q^{-1}(A)\subseteq q^{-1}(U)\text{,}\) where \(q^{-1}(A)\) is closed and \(q^{-1}(U)\) is open. Since \(X\) is normal there is an open set \(W\) satisfying \(q^{-1}(A)\subseteq W\subseteq q^{-1}(U)\text{.}\)
It is not necessarily true that \(q(W)\) is open, since we can not guarantee that \(W\) is saturated. However, since \(X-W\) is closed and \(q\) is a closed map, the image \(q(X-W)\) is closed in \(Y\text{.}\) Let \(V=Y-q(X-W)\text{.}\) Since \(q(X-W)\supseteq q(X-q^{-1}(U))=X-U\) (we use here that \(q\) is surjective), we have \(V\subseteq U\text{.}\) Since \(f^{-1}(A)\subseteq W\text{,}\) we see that \(A\cap q(X-W)=\emptyset\text{,}\) and thus \(A\subseteq V\text{.}\) We conclude that \(V\) is an open set satisfying \(A\subseteq V\subseteq U\text{,}\) showing that \(Y\) is normal.
Theorem2.18.3.Dunce cap fundamental group.
Let \(X\) be the \(n\)-fold dunces cap, let \(h\colon B^2\rightarrow X \) be the quotient map, and let \(A=h(S^1)\text{.}\)
The space \(X\) is Hausdorff and path connected, the subspace \(A\) is closed and path connected, and \(X\) is obtained from \(A\) by adjoining a 2-cell via the map \(h\text{.}\)
We have \(\pi_1(X,x_0)\cong \Z/n\Z\) for any \(x_0\in X\text{.}\)
