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Stream: theory: algebraic topology

Topic: Simplices and projective spaces

Amar Hadzihasanovic (Jan 30 2021 at 14:01):

I noticed this particularly nice construction of real projective spaces $P^n$ as quotients of the standard simplices $\Delta^n$.
Let $p_0: \Delta^0 \to P^0$ be the identity.

For $n > 0$, suppose $p_{n-1}$ is defined. There is a unique map of simplicial sets $\partial p_n: \partial \Delta^n \to P^{n-1}$ such that

• $\partial p_n \circ d^0 = \partial p_n \circ d^n = p_{n-1}$, where $d^0$ and $d^n$ are the first and last face of $\Delta^n$;
• on all other faces, the image of $\partial p_n$ is a degenerate simplex.

Then let $P^n$ be the pushout of $\partial p_n$ and the inclusion $\partial \Delta^n \hookrightarrow \Delta^n$ in simplicial sets, and $p_n: \Delta^n \to P^n$ the map obtained as part of the pushout square.

This is particularly nice because it makes the cellular homology of $P^n$ trivial to compute; the “alternating” behaviour of the homology groups is explained by the fact that the first and last face of the $n$-simplex have the same or opposite orientations depending on the parity of $n$.

Have you seen this construction before?
I tried to google but I can only find more complicated simplicial complexes that present projective spaces.