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I am having trouble with the following question from Hatcher:

Construct a $\Delta$-complex structure on $\mathbb{R}P^2$ as a quotient of a $\Delta$-complex structure on $S^n$ having vertices the two vectors of length 1 along each coordinate axis in $\mathbb{R}^{n + 1}$.

Susan
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Let me describe the case $n = 2$ (it is generalized easily).

Let $e_0, e_1, e_2$ be the canonical basis vectors of $\mathbf{R}^{3}$ and consider the $\Delta$-complex with the six vertices $\pm e_i$, and eight 2-dimensional faces $[\pm e_0, \pm e_1, \pm e_2]$, glued along the edges $[\pm e_i, \pm e_j]$ ($i \ne j$). Basically, you just have a 2-simplex in each quadrant. This is a $\Delta$-complex homeomorphic to $S^2$ and by identifying opposite faces $[\pm e_0, \pm e_1, \pm e_n] \sim [\mp e_0, \ldots, \mp e_n]$ one obtains the real projective space $\mathbf{R}\mathbf{P}^2$.

  • How would one go about showing it is homeomorphic to $S^2$? Also, do you mean eight $2$-dimensional faces? – rondo9 Apr 27 '13 at 21:44
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    Yes, sorry, one face in each quadrant. Can you see intuitively why it is homeomorphic to a sphere? Imagine "un-flattening" all the faces... –  Apr 28 '13 at 08:19