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Obstructions to almost complex structures following Massey

Michael Albanese, Aleksandar Milivojevic

TL;DR

The paper provides rigorous proofs of Massey’s two central obstructions to equipping a real vector bundle with a complex structure, using the Moore–Postnikov tower to translate lifting questions into cohomological obstructions. It identifies the first obstruction $W_3=eta w_2$ and analyzes the second obstruction, including the stable case $W_7$ and a rank-$6$ explicit formula that expresses the obstruction as a fractional component of a combination of Chern, Pontryagin, and Euler data via $4 rak{o}= extstyleigl( extstyle extsum_{i+j=2k}(-1)^i c_ic_j -(-1)^k p_kigr)$. A central achievement is Massey’s Theorem I, which relates the higher obstruction $ rak{o}$ to $W_{4k+3}$ by a scaling factor $ ext{ℓ}=(2k)!$ (even $k$) or $ frac{1}{2}(2k)!$ (odd $k$), and Massey’s Theorem II, giving explicit expressions tying the obstruction to Chern and Pontryagin data, with concrete examples on projective and quaternionic projective spaces to illustrate the sharpness and limitations of the integral obstructions. The work ties these obstructions to classical results of Wu, Ehresmann, and related K-theoretic invariants, providing a cohesive obstruction-theoretic framework for almost complex structures on manifolds and clarifying the roles of integral versus mod-$2$ phenomena through representative examples like $ frac{1}{ ext{HP}}$ and $ ext{HP}^2$.

Abstract

We provide proofs of two theorems stated by Massey in 1961, concerning the obstructions to finding complex structures on real vector bundles. In addition, we determine the second obstruction to a complex structure on a rank six orientable real vector bundle. The obstructions are fractional parts of integral Stiefel-Whitney classes, and a fourth of an appropriate combination of Pontryagin, Chern, and Euler classes.

Obstructions to almost complex structures following Massey

TL;DR

The paper provides rigorous proofs of Massey’s two central obstructions to equipping a real vector bundle with a complex structure, using the Moore–Postnikov tower to translate lifting questions into cohomological obstructions. It identifies the first obstruction and analyzes the second obstruction, including the stable case and a rank- explicit formula that expresses the obstruction as a fractional component of a combination of Chern, Pontryagin, and Euler data via . A central achievement is Massey’s Theorem I, which relates the higher obstruction to by a scaling factor (even ) or (odd ), and Massey’s Theorem II, giving explicit expressions tying the obstruction to Chern and Pontryagin data, with concrete examples on projective and quaternionic projective spaces to illustrate the sharpness and limitations of the integral obstructions. The work ties these obstructions to classical results of Wu, Ehresmann, and related K-theoretic invariants, providing a cohesive obstruction-theoretic framework for almost complex structures on manifolds and clarifying the roles of integral versus mod- phenomena through representative examples like and .

Abstract

We provide proofs of two theorems stated by Massey in 1961, concerning the obstructions to finding complex structures on real vector bundles. In addition, we determine the second obstruction to a complex structure on a rank six orientable real vector bundle. The obstructions are fractional parts of integral Stiefel-Whitney classes, and a fourth of an appropriate combination of Pontryagin, Chern, and Euler classes.
Paper Structure (12 sections, 9 theorems, 44 equations)

This paper contains 12 sections, 9 theorems, 44 equations.

Table of Contents

  1. Introduction
  2. Moore--Postnikov systems
  3. The Postnikov system of a space
  4. The Moore--Postnikov system of a map
  5. Naturality of the Moore--Postnikov system
  6. More preliminaries, and a proof of Ehresmann's theorem
  7. Some relevant homotopy groups
  8. The first obstruction to an almost complex structure
  9. The second obstruction to an almost complex structure
  10. Proof of Massey's Theorem I
  11. Proof of Massey's Theorem II
  12. Remarks on Massey's Theorem III

Key Result

Theorem 1

M61 Let $\xi \to X$ be an orientable real vector bundle of rank $2n$, whose structure group we assume has been reduced to $SO(2n)$, over a CW complex $X$. Consider the associated $SO(2n)/U(n)$ bundle $\theta$, sections of which correspond to reductions of the structure group of $\xi$ to $U(n)$. If $

Theorems & Definitions (18)

  • Theorem
  • Proposition 1
  • Lemma 2
  • proof
  • proof : Proof of \ref{['integralSW']}
  • Theorem 3
  • Theorem 4
  • Lemma 5
  • proof
  • Theorem 6
  • ...and 8 more