Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Monodromy and mapping class groups of 3-dimensional hypersurfaces

Oscar Randal-Williams

TL;DR

This work determines the image of the monodromy action from the universal family of smooth degree-$d$ hypersurfaces $X_d\subset \mathbb{CP}^4$ into the oriented mapping class group $\mathrm{MCG}_d$. The analysis centralizes on automorphisms of $\pi_3(X_d)$ that preserve the intrinsic antisymmetric form $\lambda$ and a Mon$_d$-invariant quadratic refinement $\mu$, yielding a surjective map $\mathrm{Mon}_d \to \mathrm{Aut}(\pi_3(X_d), \lambda, \mu)$ for $d\ge 3$ with kernel generated by diffeomorphisms supported in an embedded $6$-disc, identified with a subgroup of $\Theta_7$. The paper then bounds the image from above via cobordism and stable-homotopy techniques, computing the abelianization of stabilizers and demonstrating that the image sits inside a central extension by a finite group $\mathrm{K}_d$ and surjects onto $\mathrm{Aut}(\pi_3(X_d), \lambda, \mu)$; crucial torsion computations show the cokernel is controlled by a quotient $\mathrm{Coker}(\Phi)$. A comparison with $W_{g,1}$ connects these results to mapping classes of 3-folds built from $S^3\times S^3$, yielding a coherent picture of the monodromy action in terms of stable tangential structures and surgery kernels. The results illuminate when the monodromy image is large and connect it to the finite residuals coming from $\Theta_7$ and Kreck–Su’s theory, while providing explicit descriptions in terms of $\pi_3(X_d)$ and $H_3(X_d;\mathbb{Z})$.

Abstract

We describe the subgroup of the mapping class group of a hypersurface in $\mathbb{CP}^4$ consisting of those diffeomorphisms which can be realised by monodromy.

Monodromy and mapping class groups of 3-dimensional hypersurfaces

TL;DR

This work determines the image of the monodromy action from the universal family of smooth degree- hypersurfaces into the oriented mapping class group . The analysis centralizes on automorphisms of that preserve the intrinsic antisymmetric form and a Mon-invariant quadratic refinement , yielding a surjective map for with kernel generated by diffeomorphisms supported in an embedded -disc, identified with a subgroup of . The paper then bounds the image from above via cobordism and stable-homotopy techniques, computing the abelianization of stabilizers and demonstrating that the image sits inside a central extension by a finite group and surjects onto ; crucial torsion computations show the cokernel is controlled by a quotient . A comparison with connects these results to mapping classes of 3-folds built from , yielding a coherent picture of the monodromy action in terms of stable tangential structures and surgery kernels. The results illuminate when the monodromy image is large and connect it to the finite residuals coming from and Kreck–Su’s theory, while providing explicit descriptions in terms of and .

Abstract

We describe the subgroup of the mapping class group of a hypersurface in consisting of those diffeomorphisms which can be realised by monodromy.
Paper Structure (24 sections, 26 theorems, 123 equations, 2 figures)

This paper contains 24 sections, 26 theorems, 123 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Some recollections
  3. Algebraic topology of hypersurfaces
  4. Monodromy
  5. The mapping class group
  6. Upper bounds: tangential structures and quadratic forms
  7. Tangential structures
  8. The tangential structure $\theta^{\mathrm{hyp}}$
  9. Distortion of the first Pontrjagin class
  10. The surgery kernel
  11. Lower bounds: automorphisms of the Fermat hypersurface
  12. Dehn twists and the Milnor sphere
  13. Automorphisms of $\pi_3(X_d)$ realised by monodromy
  14. Further upper bounds: cobordism considerations
  15. Calculating the abelianisation of $\mathrm{Stab}_{\mathrm{MCG}_d}(\ell^{\mathrm{hyp}}_{X_d})$
  16. ...and 9 more sections

Key Result

Theorem A

Suppose $d \geq 3$.

Figures (2)

  • Figure 1: $E_2$-page of the Adams spectral sequence at $p=3$ for $MT\bar{\theta}^\mathrm{hyp}$, with $d \equiv 0 \mod 3$.
  • Figure 2: $E_2$-page of the Adams spectral sequence at $p=2$ for $MT\bar{\theta}^\mathrm{hyp}$, with $d$ even. The class of filtration 1 in degree 7 detects the image of $\sigma \in \pi_7^s(S^0)$ on the bottom cell $\iota : S^0 \to MT\bar{\theta}^\mathrm{hyp}$.

Theorems & Definitions (54)

  • Theorem A
  • Theorem B
  • Remark 1.1
  • Theorem 2.1: Beauville Beauville
  • Lemma 2.2: Kreck--Su KSv3
  • proof : Proof sketch
  • Remark 3.1
  • Lemma 3.2
  • proof
  • Remark 3.3
  • ...and 44 more