Algebraic & Geometric Topology

Rational analogs of projective planes

Zhixu Su

Full-text: Open access


In this paper, we study the existence of high-dimensional, closed, smooth manifolds whose rational homotopy type resembles that of a projective plane. Applying rational surgery, the problem can be reduced to finding possible Pontryagin numbers satisfying the Hirzebruch signature formula and a set of congruence relations, which turns out to be equivalent to finding solutions to a system of Diophantine equations.

Article information

Algebr. Geom. Topol., Volume 14, Number 1 (2014), 421-438.

Received: 15 October 2010
Revised: 22 July 2013
Accepted: 22 July 2013
First available in Project Euclid: 19 December 2017

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 57R20: Characteristic classes and numbers
Secondary: 57R65: Surgery and handlebodies 57R67: Surgery obstructions, Wall groups [See also 19J25]

rational surgery rational homotopy type smooth manifold


Su, Zhixu. Rational analogs of projective planes. Algebr. Geom. Topol. 14 (2014), no. 1, 421--438. doi:10.2140/agt.2014.14.421.

Export citation


  • D R Anderson, On homotopy spheres bounding highly connected manifolds, Trans. Amer. Math. Soc. 139 (1969) 155–161
  • J Barge, Structures différentiables sur les types d'homotopie rationnelle simplement connexes, Ann. Sci. École Norm. Sup. 9 (1976) 469–501
  • I H Madsen, R J Milgram, The classifying spaces for surgery and cobordism of manifolds, Annals of Mathematics Studies 92, Princeton Univ. Press (1979)
  • J Milnor, Collected papers of John Milnor, III: Differential topology, Amer. Math. Soc. (2007)
  • J Milnor, D Husemoller, Symmetric bilinear forms, Ergeb. Math. Grenzgeb. 73, Springer, Berlin (1973)
  • J W Milnor, J D Stasheff, Characteristic classes, Annals of Mathematics Studies 76, Princeton Univ. Press (1974)
  • R E Stong, Relations among characteristic numbers, I, Topology 4 (1965) 267–281
  • R E Stong, Notes on cobordism theory, Princeton Univ. Press (1968)
  • Z Su, Rational homotopy type of manifolds, PhD thesis, Indiana University (2009) Available at \setbox0\makeatletter\@url {\unhbox0
  • D Sullivan, Infinitesimal computations in topology, Inst. Hautes Études Sci. Publ. Math. (1977) 269–331
  • L Taylor, B Williams, Local surgery: Foundations and applications, from: “Algebraic topology”, (J L Dupont, I H Madsen, editors), Lecture Notes in Math. 763, Springer, Berlin (1979) 673–695