Closed manifold surgery obstructions and the Oozing Conjecture

TL;DR

The article delivers a complete description of surgery obstructions up to homotopy for closed oriented manifolds with finite fundamental group, and it exposes new nontrivial Arf‑invariant obstructions in codimensions $\ge 4$, providing counterexamples to the Oozing Conjecture. It develops a 2‑adic lifting framework via the quadratic extension ring $R=\mathbb Z[\varepsilon]$, derives explicit universal homomorphisms $\kappa^s_j$ and $\kappa'^s_j$, and connects these to group‑homology through the assembly map (Theorem E). The paper gives sharp results for abelian/basic $2$‑groups (Theorem A) and analyzes the images under coefficient change (Theorems B–C), while showing ker$s$ vs ker$h$ gaps (Corollary D) and constructing concrete nonvanishing $\,\kappa'_4$ examples (Theorem/Corollary ooze). It also demonstrates the failure of the oozing conjecture in the simple‑homotopy setting by reducing to explicit group‑theory computations and providing codimension‑two twisting constructions that separate simple from non-simple obstructions. Collectively, these contributions illuminate the structure of surgery obstructions, advance the understanding of Arf invariant product formulas, and supply explicit finite‑group examples that sharpen the landscape of higher codimension obstructions.

Abstract

We complete the description of surgery obstructions up to homotopy equivalence for closed oriented manifolds with finite fundamental group. New examples are presented of non-trivial obstructions for Arf invariant product formulas in codimensions $\geq 4$, which give counterexamples to the well-known ''Oozing Conjecture'' from the 1980's.

Theorems & Definitions (67)

• Remark 1.1
• Remark 1.2
• Remark 1.3
• Remark 1.4
• Corollary 1.5
• Remark 1.6
• proof : The proof of Theorem A
• Theorem 3.1: Milgram-Oliver Milgram:1990
• Corollary 3.2
• Lemma 3.3