User:Harold/Cone of rp2

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

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

We now compute [ilmath] H_1(C(\rp2), U) [/ilmath]. We have [ilmath] H_0(C(\rp2)) \cong 0 \cong H_0(U) [/ilmath]. We have the relevant portion of the long exact sequence for pairs, [math] \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. [/math]

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

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

[math] \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. [/math]

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

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