Geodesic
Generalised Kummer construction and the cohomology rings of $G_2$-manifolds
Sbornik. Mathematics, Tome 209 (2018) no. 12, pp. 1803-1811 Cet article a éte moissonné depuis la source Math-Net.Ru

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Intersection theory is used to calculate the cohomology rings of $G_2$-manifolds arising from the generalised Kummer construction. For one example, generators of the rational cohomology ring are found and their multiplication table is described. Bibliography: 19 titles.
Keywords: cohomology ring, intersection ring, manifolds with $G_2$-holonomy. 
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I. A. Taimanov. Generalised Kummer construction and the cohomology rings of $G_2$-manifolds. Sbornik. Mathematics, Tome 209 (2018) no. 12, pp. 1803-1811. http://geodesic.mathdoc.fr/item/SM_2018_209_12_a6/

[1] D. D. Joyce, “Compact Riemannian $7$-manifolds with holonomy $G_2$. I”, J. Differential Geom., 43:2 (1996), 291–328 | DOI | MR | Zbl

[2] D. D. Joyce, “Compact Riemannian $7$-manifolds with holonomy $G_2$. II”, J. Differential Geom., 43:2 (1996), 329–375 | DOI | MR | Zbl

[3] D. D. Joyce, “Compact $8$-manifolds with holonomy $\operatorname{Spin}(7)$”, Invent. Math., 123:3 (1996), 507–552 | DOI | MR | Zbl

[4] D. D. Joyce, Compact manifolds with special holonomy, Oxford Math. Monogr., Oxford Univ. Press, Oxford, 2000, xii+436 pp. | MR | Zbl

[5] D. Joyce, “A new construction of compact $8$-manifolds with holonomy $\operatorname{Spin}(7)$”, J. Differential Geom., 53:1 (1999), 89–130 | DOI | MR | Zbl

[6] A. Kovalev, “Twisted connected sums and special Riemannian holonomy”, J. Reine Angew. Math., 2003:565 (2003), 125–160 | DOI | MR | Zbl

[7] A. Kovalev, Nam-Hoon Lee, “$K3$-surfaces with non-symplectic involution and compact irreducible $G_2$-manifolds”, Math. Proc. Cambridge Philos. Soc., 151:2 (2011), 193–218 | DOI | MR | Zbl

[8] A. Corti, M. Haskins, J. Nordström, T. Pacini, “$G_2$-manifolds and associative submanifolds via semi-Fano $3$-folds”, Duke Math. J., 164:10 (2015), 1971–2092 | DOI | MR | Zbl

[9] J. W. Milnor, “On simply connected $4$-manifolds”, Symposium internacional de topología algebraica, Universidad Nacional Autónoma de México and UNESCO, Mexico City, 1958, 122–128 | MR | Zbl

[10] I. A. Taimanov, “A canonical basis of two-dimensional cycles on a $K3$ surface”, Mat. Sb., 209:8 (2018), 1248–1256 | DOI | DOI | MR

[11] M. E. Glezerman, L. S. Pontryagin, “Peresecheniya v mnogoobraziyakh”, UMN, 2:1(17) (1947), 58–155 | MR

[12] I. R. S̆afarevic̆, B. G. Averbuh, Ju. R. Vaĭnberg, A. B. Z̆iz̆c̆enko, Ju. I. Manin, B. G. Moĭs̆ezon, G. N. Tjurina, A. N. Tjurin, “Algebraic surfaces”, Proc. Steklov Inst. Math., 75 (1965), 1–281 | MR | MR | Zbl | Zbl

[13] D. Huybrechts, Lectures on K3 surfaces, Cambridge Stud. Adv. Math., 158, Cambridge Univ. Press, Cambridge, 2016, xi+485 pp. | DOI | MR | Zbl

[14] P. Deligne, P. Griffits, J. Morgan, D. Sullivan, “Real homotopy theory of Kähler manifolds”, Invent. Math., 29:3 (1975), 245–274 | DOI | MR | MR | Zbl | Zbl

[15] D. Sullivan, “Infinitesimal computations in topology”, Inst. Hautes Études Sci. Publ. Math., 47 (1977), 269–331 | DOI | MR | Zbl

[16] I. K. Babenko, I. A. Taĭmanov, “On nonformal simply-connected symplectic manifolds”, Siberian Math. J., 41:2 (2000), 204–217 | DOI | MR | Zbl

[17] I. K. Babenko, I. A. Taimanov, “Massey products in symplectic manifolds”, Sb. Math., 191:8 (2000), 1107–1146 | DOI | DOI | MR | Zbl

[18] M. Fernández, V. Muñoz, “An $8$-dimensional nonformal, simply connected, symplectic manifold”, Ann. of Math. (2), 167:3 (2008), 1045–1054 | DOI | MR | Zbl

[19] M. Fernández, S. Ivanov, V. Muñoz, Formality of $7$-dimensional $3$-Sasakian manifolds, 2015, arXiv: 1511.08930