User:Harold/Cone of rp2

$\newcommand{\rp}[1]{\mathbb{R}P^{#1}} \newcommand{\R}{\mathbb{R} }$ Assume that $C(\rp2)$ is a topological 3-manifold with boundary. Let $x$ be an interior point (i.e., not a boundary point) of $C(\rp^2)$. We compute $H_1(C(\rp2) - \{ x \}$.

Let be a small open neighbourhood of $x$ that is homeomorphic to an open $\epsilon$-ball around the origin in $\R^3$. The subsets $x \subset U \subset C(\rp2)$ satisfy the conditions of Hatcher, Theorem 2.20 (the Excision Theorem). Hence we obtain $H_n(C(\rp2) - \{ x \}, U - \{ x \}) \cong H_n(C(\rp2), U)$ for all $n \in \mathbb{Z}$.

We now compute $H_1(C(\rp2), U)$. We have $H_0(C(\rp2)) \cong 0 \cong H_0(U)$. We have the relevant portion of the long exact sequence for pairs, $$\to \underbrace{H_1(U)}_{\cong 0} \to \underbrace{H_1(C(\rp2))}_{\cong 0} \to H_1(C(\rp2), U) \to \underbrace{H_0(U)}_{\cong \mathbb{Z} } \to \underbrace{H_0(C(\rp2))}_{\cong \mathbb{Z} } \to H_0(C(\rp2), U) \to 0.$$

From the exactness of this sequence, we conclude that $H_1(C(\rp2), U)$ is either isomorphic to $\mathbb{Z}$ or is trivial.

Looking at the long exact sequence for the pair $(C(\rp2) - \{ x \} , U - \{ x \})$, we see

$$\to \underbrace{H_1(U - \{ x \})}_{\cong 0} \to H_1(C(\rp2) - \{ x \} ) \to \underbrace{H_1(C(\rp2) - \{x\}, U - \{x\})}_{\cong H_1(C(\rp2), U)} \to \underbrace{H_0(U - \{x\})}_{\cong \mathbb{Z} } \to \underbrace{H_0(C(\rp2) - \{x\})}_{\cong \mathbb{Z} } \to H_0(C(\rp2) - \{x\}, U - \{x\}) \to 0.$$

By exactness, we obtain that the map $H_1(C(\rp2) - \{ x \}) \to H_1(C(\rp2) - \{x\}, U - \{x\})$ is injective. As we previously concluded, $H_1(C(\rp2), U)$ is either isomorphic to $\mathbb{Z}$ or is trivial, hence $H_1(C(\rp2) - \{ x \})$ is either $\mathbb{Z}$ or trivial.

Now let $p$ be the cone point of $C(\rp2)$. Then $C(\rp2) - \{ p \}$ is homotopy equivalent to $\rp2$, hence $H_1(C(\rp2) - \{p\}$ is isomorphic to $\mathbb{Z} / 2 \mathbb{Z}$. Hence $p$ cannot be an interior point, nor a boundary point of $C(\rp2)$, and as such, $C(\rp2)$ cannot be a topological manifold with boundary.