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The smooth classification of $4$-dimensional complete intersections

Diarmuid Crowley and Csaba Nagy

Geometry & Topology 29 (2025) 269–311
Abstract

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 Pn+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 Θ82.

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 S1 and the Sullivan conjecture.

Keywords
complete intersection, diffeomorphism classification, Sullivan conjecture, degree-d normal map
Mathematical Subject Classification
Primary: 57R55
Secondary: 32J18
References
Publication
Received: 26 March 2020
Revised: 11 December 2023
Accepted: 28 January 2024
Published: 22 January 2025
Proposed: András I Stipsicz
Seconded: Ciprian Manolescu, Benson Farb
Authors
Diarmuid Crowley
School of Mathematics and Statistics
University of Melbourne
Parkville, VIC
Australia
Csaba Nagy
School of Mathematics and Statistics
University of Glasgow
Glasgow
United Kingdom

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