Geodesic
Rational homology of the order complex of zero sets of homogeneous quadratic polynomial systems in~$\mathbb R^3$
Informatics and Automation, Modern problems of mathematics, mechanics, and mathematical physics, Tome 290 (2015), pp. 211-225

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The naturally topologized order complex of proper algebraic subsets in $\mathbb R\mathrm P^2$ defined by systems of quadratic forms has the rational homology of $S^{13}$. 
@article{TRSPY_2015_290_a17,
     author = {V. A. Vassiliev},
     title = {Rational homology of the order complex of zero sets of homogeneous quadratic polynomial systems in~$\mathbb R^3$},
     journal = {Informatics and Automation},
     pages = {211--225},
     publisher = {mathdoc},
     volume = {290},
     year = {2015},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TRSPY_2015_290_a17/}
}
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V. A. Vassiliev. Rational homology of the order complex of zero sets of homogeneous quadratic polynomial systems in~$\mathbb R^3$. Informatics and Automation, Modern problems of mathematics, mechanics, and mathematical physics, Tome 290 (2015), pp. 211-225. http://geodesic.mathdoc.fr/item/TRSPY_2015_290_a17/