From the icosahedron to natural triangulations of $\CC P^2$ and $S^2 \times S^2$

Details From the icosahedron to natural triangulations of $\CC P^2$ and $S^2 \times S^2$ From the icosahedron to natural triangulations of $\CC P^2$ and $S^2 \times S^2$

2010-04-19

arxiv.org/abs/1004.3157v2

Discrete Comput Geom 46 (2011), 542--560

Mathematics

Algebraic Topology

18 pages, revised version, to appear in `Discrete & Computational Geometry'

Scientific paper

We present two constructions in this paper: (a) A 10-vertex triangulation $\CC P^{2}_{10}$ of the complex projective plane $\CC P^{2}$ as a subcomplex of the join of the standard sphere ($S^{2}_4$) and the standard real projective plane ($\RR P^{2}_{6}$, the decahedron), its automorphism group is $A_4$; (b) a 12-vertex triangulation $(S^{2} \times S^{2})_{12}$ of $S^{2} \times S^{2}$ with automorphism group $2S_5$, the Schur double cover of the symmetric group $S_5$. It is obtained by generalized bistellar moves from a simplicial subdivision of the standard cell structure of $S^{2} \times S^{2}$. Both constructions have surprising and intimate relationships with the icosahedron. It is well known that $\CC P^{2}$ has $S^{2} \times S^{2}$ as a two-fold branched cover; we construct the triangulation $\CC P^{2}_{10}$ of $\CC P^{2}$ by presenting a simplicial realization of this covering map $S^{2} \times S^{2} \to \CC P^{2}$. The domain of this simplicial map is a simplicial subdivision of the standard cell structure of $S^{2} \times S^{2}$, different from the triangulation alluded to in (b). This gives a new proof that Kuehnel's $\CC P^{2}_{9}$ triangulates $\CC P^{2}$. It is also shown that $\CC P^{2}_{10}$ and $(S^{2} \times S^{2})_{12}$ induce the standard piecewise linear structure on $\CC P^{2}$ and $S^{2} \times S^{2}$ respectively.

Affiliated with

Also associated with

No associations

Rating

From the icosahedron to natural triangulations of $\CC P^2$ and $S^2 \times S^2$ does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with From the icosahedron to natural triangulations of $\CC P^2$ and $S^2 \times S^2$, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and From the icosahedron to natural triangulations of $\CC P^2$ and $S^2 \times S^2$ will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-59605