A 15-vertex triangulation of the quaternionic projective plane. (English) Zbl 1425.57016

U. Brehm and W. Kühnel [Math. Ann. 294, No. 1, 167–193 (1992; Zbl 0734.57017)] found a triangulated 8-manifold \(M^8_{15}\) with 15 vertices which is not a sphere. They gave a heuristic argument supporting that \(M^8_{15}\) is homeomorphic with the quaternionic projective plane, but that was not a proof. From the decomposition into 9:6 vertices (8-simplex vs. boundary of 5-simplex) it was clear that it is a triangulated manifold “like a projective plane” in the sense by J. Eells jun. and N. H. Kuiper [Publ. Math., Inst. Hautes Étud. Sci. 14, 181–222 (1962; Zbl 0109.15701)] and L. Kramer [J. Differ. Geom. 64, No. 1, 1–55 (2003; Zbl 1068.57019)]. The Hirzebruch formula for the signature implies that it is sufficient to compute the first or second Pontryagin class but that was out of range for a long time, even though theoretically there was a formula by Gabrielov-Gelfand-Losik. But that could not practically be evaluated.
In the remarkable paper under review the author manages to compute the first Pontryagin class of \(M^8_{15}\) in a purely combinatorial way. This is based on previous work by A. A. Gaifullin [Izv. Math. 68, No. 5, 861–910 (2004; Zbl 1068.57022); translation from Izv. Ross. Akad. Nauk Ser. Mat. 68, No. 5, 13–66 (2004)]. Roughly speaking Gaifullin expressed the first Pontryagin class in terms of bistellar moves between all 3-dimensional links in the triangulated 8-manifold. This defines cycles in the graph \(\Gamma_2\) consisting of all oriented 2-spheres as vertices and bistellar moves as edges. Since \(M^8_{15}\) has \(\binom{15}{5}= 3003\) 4-simplices (with more than 60 combinatorial types even after regarding the automorphism group) one has to examine as many 3-dimensional links. Ultimately the calculation depends on a computer algorithm. This in turn is based on the mathematical software BISTELLAR which is available from the homepage of Frank H. Lutz.


57R20 Characteristic classes and numbers in differential topology
57Q15 Triangulating manifolds
52B70 Polyhedral manifolds
55R40 Homology of classifying spaces and characteristic classes in algebraic topology
57N65 Algebraic topology of manifolds


simpcomp; GAP
Full Text: DOI arXiv


[1] Björner, A.; Lutz, FH, Simplicial manifolds, bistellar flips and a 16-vertex triangulation of the Poincaré homology 3-sphere, Exp. Math., 9, 275-289, (2000) · Zbl 1101.57306
[2] Brehm, U.; Kühnel, W., Combinatorial manifolds with few vertices, Topology, 26, 465-473, (1987) · Zbl 0681.57009
[3] Brehm, U.; Kühnel, W., 15-Vertex triangulations of 8-manifolds, Math. Ann., 294, 167-193, (1992) · Zbl 0734.57017
[4] Brumfiel, G., On integral PL characteristic classes, Topology, 8, 39-46, (1969) · Zbl 0172.25301
[5] Cheeger, J., Spectral geometry of singular Riemannian spaces, J. Differ. Geom., 18, 575-657, (1983) · Zbl 0529.58034
[6] Eells, J.; Kuiper, NH, Manifolds which are like projective planes, Inst. Hautes Études Sci. Publ. Math., 14, 5-46, (1962) · Zbl 0109.15701
[7] Effenberger, F., Spreer, J.: simpcomp—a GAP toolkit for simplicial complexes, Version 2.0.0 (2013). http://code.google.com/p/simpcomp · Zbl 1308.68167
[8] Forman, R., Morse theory for cell complexes, Adv. Math., 134, 90-145, (1998) · Zbl 0896.57023
[9] Gabrièlov, AM; Gel’fand, IM; Losik, MV, Combinatorial calculation of characteristic classes, Funct. Anal. Appl., 9, 48-49, (1975) · Zbl 0312.57015
[10] Gabrièlov, AM; Gel’fand, IM; Losik, MV, Combinatorial calculus of characteristic classes, Funct. Anal. Appl., 9, 186-202, (1975) · Zbl 0341.57017
[11] Gabrièlov, AM; Gel’fand, IM; Losik, MV, A local combinatorial formula for the first class of Pontryagin, Funct. Anal. Appl., 10, 12-15, (1976) · Zbl 0328.57006
[12] Gaifullin, AA, Local formulae for combinatorial Pontryagin classes, Izv. Math., 68, 861-910, (2004) · Zbl 1068.57022
[13] Gaifullin, AA, The construction of combinatorial manifolds with prescribed sets of links of vertices, Izv. Math., 72, 845-899, (2008) · Zbl 1156.52009
[14] Gaifullin, AA, Configuration spaces, bistellar moves, and combinatorial formulae for the first Pontryagin class, Proc. Steklov Inst. Math., 268, 70-86, (2010) · Zbl 1227.57033
[15] Gel’fand, IM; MacPherson, RD, A combinatorial formula for the Pontrjagin classes, Bull. Am. Math. Soc., 26, 304-309, (1992) · Zbl 0756.57015
[16] Gorodkov, D., A minimal triangulation of the quaternionic projective plane, Russ. Math. Surv., 71, 1140-1142, (2016) · Zbl 1369.57027
[17] Kervaire, MA; Milnor, JW, Groups of homotopy spheres: I, Ann. Math., 77, 504-537, (1963) · Zbl 0115.40505
[18] Kramer, L., Projective planes and their look-alikes, J. Differ. Geom., 64, 1-55, (2003) · Zbl 1068.57019
[19] Lutz, F.H.: Triangulated manifolds with few vertices: Combinatorial manifolds (2005). arxiv:0506372
[20] MacPherson, R.: The combinatorial formula of Gabrielov, Gel’fand and Losik for the first Pontrjagin class. Lecture Notes in Mathematics, vol. 677, pp. 105-124. Springer, Berlin (1978) · Zbl 0388.57013
[21] Milin, L., A combinatorial computation of the first Pontryagin class of the complex projective plane, Geom. Dedicata, 49, 253-291, (1994) · Zbl 0956.57016
[22] Pachner, U., Konstruktionsmethoden und das kombinatorische homöomorphieproblem für triangulationen kompakter semilinearer mannigfaltigkeiten, Abh. Math. Sem. Univ. Hamb., 57, 69-86, (1987) · Zbl 0651.52007
[23] The GAP Group. GAP—Groups, Algorithms, and Programming, Version 4.7.7 (2015). http://www.gap-system.org
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.