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The smooth classification of 4-dimensional complete intersections

Diarmuid Crowley, Csaba Nagy

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 $CP^{n+k}$. 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 $Θ_8 = Z/2$. 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.

The smooth classification of 4-dimensional complete intersections

Abstract

We prove the "Sullivan Conjecture" on the classification of 4-dimensional complete intersections up to diffeomorphism. Here an -dimensional complete intersection is a smooth complex variety formed by the transverse intersection of hypersurfaces in . 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 . Our proof involves several new ideas, including the use of the Hambleton-Madsen theory of degree- 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 and the Sullivan Conjecture.

Paper Structure

This paper contains 21 sections, 45 theorems, 81 equations.

Table of Contents

  1. Introduction
  2. Complete intersections and the Sullivan Conjecture
  3. Background and an application
  4. The outline of the proof of Theorem \ref{['thm:main']}
  5. Inertia groups of $4$-dimensional complete intersections
  6. Preliminaries
  7. Complete intersections
  8. Computation of Sullivan data and the converse of the Sullivan Conjecture
  9. The setting for modified surgery
  10. The spin case
  11. The non-spin cases with even total degree
  12. The computation of $\mathop{\mathrm{Tors}}\nolimits \Omega^{O\langle{7}\rangle}_8(\mathbb{C} P^1; \xi^1)$
  13. The non-spin case with $v_4(X_4(\ul d)) = 0$
  14. The non-spin case with $v_4(X_4(\ul d)) \neq 0$ and even total degree
  15. The case of odd total degree
  16. ...and 6 more sections

Key Result

Theorem 1.3

Suppose that $X_4(\ul d)$ and $X_4(\ul d')$ are complete intersections with $SD_4(\ul d) = SD_4(\ul d')$. Then $X_4(\ul d)$ is diffeomorphic to $X_4(\ul d')$.

Theorems & Definitions (108)

  • Definition 1.1
  • Conjecture 1.2: The Sullivan Conjecture
  • Theorem 1.3
  • Theorem 1.4: Fang and Klaus F-K
  • Example 1.5
  • Definition 1.6
  • Theorem 1.7
  • Remark 1.8
  • Proposition 1.9
  • Remark 1.10
  • ...and 98 more