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Normal-Euler excess for disjoint nonorientable surfaces in a closed $4$-manifold

Bennett Chow, Michael Freedman

Abstract

Let \(M\) be a closed connected oriented topological \(4\)-manifold. We prove that if \(F_1,\dots,F_r\subset M\) are pairwise disjoint connected locally flat topologically embedded nonorientable surfaces with nonorientable genera \(g_i\), same-sign twisted normal Euler numbers \(e_i\), and \( [F_1]+\cdots+[F_r]=0\in H_2(M;\F_2), \) then the normal-Euler excess \( \sum_{i=1}^r \bigl(\abs{e_i}-2g_i\bigr) \) is bounded above by a constant depending only on \(M\). Thus same-sign mod-\(2\)-null families of disjoint nonorientable surfaces in a fixed ambient \(4\)-manifold have uniformly bounded total excess over Massey's \(S^4\) bound. The proof combines a tubing construction with the signature and Euler-characteristic formulas for \(2\)-fold branched covers. As corollaries, every closed oriented topological \(4\)-manifold contains only finitely many pairwise disjoint locally flat topologically embedded copies of \(\RP^2\) with \(\abs{e}>2\), and only finitely many pairwise disjoint tubular neighborhoods modeled on real \(2\)-plane bundles over \(\RP^2\) whose total spaces are orientable and whose twisted Euler numbers have absolute value greater than \(2\). When \(M\) is a homology \(4\)-sphere, the ambient error term vanishes, and the theorem recovers Massey's sharp inequality \(\abs{e(F)}\le 2g(F)\) for nonorientable surfaces in \(S^4\).

Normal-Euler excess for disjoint nonorientable surfaces in a closed $4$-manifold

Abstract

Let be a closed connected oriented topological -manifold. We prove that if are pairwise disjoint connected locally flat topologically embedded nonorientable surfaces with nonorientable genera , same-sign twisted normal Euler numbers , and \( [F_1]+\cdots+[F_r]=0\in H_2(M;\F_2), \) then the normal-Euler excess \( \sum_{i=1}^r \bigl(\abs{e_i}-2g_i\bigr) \) is bounded above by a constant depending only on . Thus same-sign mod--null families of disjoint nonorientable surfaces in a fixed ambient -manifold have uniformly bounded total excess over Massey's bound. The proof combines a tubing construction with the signature and Euler-characteristic formulas for -fold branched covers. As corollaries, every closed oriented topological -manifold contains only finitely many pairwise disjoint locally flat topologically embedded copies of with , and only finitely many pairwise disjoint tubular neighborhoods modeled on real -plane bundles over whose total spaces are orientable and whose twisted Euler numbers have absolute value greater than . When is a homology -sphere, the ambient error term vanishes, and the theorem recovers Massey's sharp inequality \(\abs{e(F)}\le 2g(F)\) for nonorientable surfaces in .

Paper Structure

This paper contains 8 sections, 7 theorems, 131 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. A linear-algebra lemma
  4. Tubing disjoint surfaces
  5. Branched covers and signatures
  6. Proof of the main theorem
  7. Projective planes and plane bundles
  8. Final remarks

Key Result

Theorem 3

Let $M$ be a closed connected oriented topological $4$-manifold. Let be pairwise disjoint connected locally flat topologically embedded nonorientable surfaces. For each $i$, let $g_i$ be the nonorientable genus of $F_i$, so that $\chi(F_i)=2-g_i$, and let be the twisted normal Euler numbers. Assume that the integers $e_1,\dots,e_r$ all have the same sign and that Then where $\sigma(M)$ denotes

Theorems & Definitions (18)

  • Definition 1
  • Remark 2
  • Theorem 3
  • Corollary 4
  • Corollary 5
  • Lemma 6
  • proof
  • Lemma 7
  • proof
  • Proposition 8
  • ...and 8 more