On Poincaré Surgery

TL;DR

The paper develops a manifold-free, homotopy-theoretic approach to Poincaré surgery in the simply connected case, introducing a surgery obstruction $\sigma(f)\in L_d(\pi,w)$ for normal maps and proving a Fundamental Theorem: if $\sigma(f)=0$ and $d\ge7$ or $d=5$, then $f$ is normally cobordant to a homotopy equivalence (with a stronger, even-dimensional Wall realization result). It builds a robust bordism framework $\mathcal{F}_d(X,\xi)$ linking normal maps to Poincaré data via a lifting map $t$, and proves surjectivity (and in some settings injectivity) results that realize obstructions as algebraic $L$-theory classes. The work develops a comprehensive Poincaré surgery toolbox, including Poincaré embeddings, middle-dimension surgery, and a non-smoothable, algebraic analogue of Wall’s realization in the even-dimensional setting, while grounding the theory in Ranicki’s algebraic surgery and the Spivak normal fibration. It also sketches the theory of Poincaré thickenings and transversality, setting the stage for extending these results beyond the simply connected case in future work.

Abstract

We exhibit a homotopy theoretic proof of the Fundamental Theorem of Poincaré surgery in the simply connected case. We also deduce the Poincaré transversality exact sequence.

Figures (4)

• Figure 1: A depiction of the Poincaré normal map constructed in the proof of Theorem \ref{['bigthm:sigma-onto']}. The white region $\bar{K} \amalg Q_0$ is mapped using the identity. On the gray region, the map supplied by Theorem \ref{['thm:Wall_realization']} is used.
• Figure 2: Poincaré embedding of $(\bar{Y},\partial \bar{Y})$ in $(P,\partial P)$.
• Figure 3: Schematic of the Poincaré cobordism $(T,T_+)$.
• Figure 4: Schematic of a codimension zero Poincaré embedding of $(P,\partial P)$ in $(M,\partial M)$.

Theorems & Definitions (63)

• Theorem 1: "Fundamental Theorem"
• Theorem 2: "Wall Realization"
• Example 1.1
• Theorem 3
• Lemma 3.1: cf. HV
• proof
• Proposition 3.2
• proof
• Remark 4.1
• Definition 5.1