The smooth classification of 4-dimensional complete intersections

Crowley, Diarmuid¹ ; Nagy, Csaba²

¹ School of Mathematics and Statistics, University of Melbourne, Parkville, VIC, Australia
² School of Mathematics and Statistics, University of Glasgow, Glasgow, United Kingdom

Geometry & topology, Tome 29 (2025) no. 1, pp. 269-311.

Résumé

We prove the “Sullivan conjecture” on the classification of $4$-dimensional complete intersections up to diffeomorphism. Here an $n$-dimensional complete intersection is a smooth complex variety formed by the transverse intersection of $k$ hypersurfaces in $ℂP^{n+k}$.

Previously Kreck and Traving proved the $4$-dimensional Sullivan conjecture when 64 divides the total degree (the product of the degrees of the defining hypersurfaces) and Fang and Klaus proved that the conjecture holds up to the action of the group of homotopy $8$-spheres ${\Theta }_8\cong \mathbb{Z}∕2$.

Our proof involves several new ideas, including the use of the Hambleton–Madsen theory of degree-$d$ normal maps, which provide a fresh perspective on the Sullivan conjecture in all dimensions. This leads to an unexpected connection between the Segal conjecture for $S^1$ and the Sullivan conjecture.

Détail

DOI : 10.2140/gt.2025.29.269

Keywords: complete intersection, diffeomorphism classification, Sullivan conjecture, degree-d normal map

Bibliographie

[1] J F Adams, On the groups J(X), IV, Topology 5 (1966) 21 | DOI

[2] D Baraglia, The alpha invariant of complete intersections, preprint (2020)

[3] W Browder, Surgery on simply-connected manifolds, 65, Springer (1972) | DOI

[4] W Browder, Complete intersections and the Kervaire invariant, from: "Algebraic topology" (editors J L Dupont, I H Madsen), Lecture Notes in Math. 763, Springer (1979) 88 | DOI

[5] E H Brown Jr., Cohomology theories, Ann. of Math. 75 (1962) 467 | DOI

[6] G Brumfiel, I Madsen, Evaluation of the transfer and the universal surgery classes, Invent. Math. 32 (1976) 133 | DOI

[7] A Dimca, Singularities and topology of hypersurfaces, Springer (1992) | DOI

[8] F Fang, S Klaus, Topological classification of 4-dimensional complete intersections, Manuscripta Math. 90 (1996) 139 | DOI

[9] F Fang, J Wang, Homeomorphism classification of complex projective complete intersections of dimensions 5, 6 and 7, Math. Z. 266 (2010) 719 | DOI

[10] M Feshbach, Essential maps exist from BU to coker J, Proc. Amer. Math. Soc. 97 (1986) 539 | DOI

[11] M Feshbach, The Segal conjecture for compact Lie groups, Topology 26 (1987) 1 | DOI

[12] M H Freedman, The topology of four-dimensional manifolds, J. Differential Geom. 17 (1982) 357

[13] V Giambalvo, On ⟨8⟩-cobordism, Illinois J. Math. 15 (1971) 533

[14] I Hambleton, I Madsen, Local surgery obstructions and space forms, Math. Z. 193 (1986) 191 | DOI

[15] A Hatcher, Algebraic topology, Cambridge University Press (2002)

[16] P J Hilton, On the homotopy groups of the union of spheres, J. Lond. Math. Soc. 30 (1955) 154 | DOI

[17] P E Jupp, Classification of certain 6-manifolds, Proc. Cambridge Philos. Soc. 73 (1973) 293 | DOI

[18] R Kasilingam, Classification of smooth structures on a homotopy complex projective space, Proc. Indian Acad. Sci. Math. Sci. 126 (2016) 277 | DOI

[19] M A Kervaire, J W Milnor, Groups of homotopy spheres, I, Ann. of Math. 77 (1963) 504 | DOI

[20] M Kreck, Surgery and duality, Ann. of Math. 149 (1999) 707 | DOI

[21] A S Libgober, J W Wood, Differentiable structures on complete intersections, I, Topology 21 (1982) 469 | DOI

[22] I Madsen, R J Milgram, The classifying spaces for surgery and cobordism of manifolds, 92, Princeton Univ. Press (1979)

[23] J Milnor, On the Whitehead homomorphism J, Bull. Amer. Math. Soc. 64 (1958) 79 | DOI

[24] J W Milnor, J D Stasheff, Characteristic classes, 76, Princeton Univ. Press (1974)

[25] R E Mosher, M C Tangora, Cohomology operations and applications in homotopy theory, Harper Row (1968)

[26] C Nagy, The classification of 8-dimensional E-manifolds, PhD thesis, University of Melbourne (2021)

[27] D C Ravenel, The Segal conjecture for cyclic groups and its consequences, Amer. J. Math. 106 (1984) 415 | DOI

[28] B J Sanderson, Immersions and embeddings of projective spaces, Proc. Lond. Math. Soc. 14 (1964) 137 | DOI

[29] S Smale, On the structure of manifolds, Amer. J. Math. 84 (1962) 387 | DOI

[30] D P Sullivan, Geometric topology : localization, periodicity and Galois symmetry, 8, Springer (1970)

[31] H Toda, Composition methods in homotopy groups of spheres, 49, Princeton Univ. Press (1962)

[32] C T C Wall, Classification problems in differential topology, V : On certain 6-manifolds, Invent. Math. 1 (1966) 355 | DOI

[33] C T C Wall, Surgery on compact manifolds, 1, Academic Press (1970)

[34] G W Whitehead, Elements of homotopy theory, 61, Springer (1978) | DOI