Main Question and Subtlety
The folk wisdom, often cited as a clear truth, states, "PL = SMOOTH in dimension 4". While intuitively appealing, this is not a precise mathematical statement. The actual relationship is subtler and better formulated in terms of structures:
- For any smooth manifold, at least one compatible PL-structure exists.
- In dimension 4, any PL-manifold also admits at least one compatible smooth structure.
The central, more formal question, then, is: Given a closed, oriented PL 4-manifold, how many compatible smooth structures exist, up to a suitable equivalence? My impression, gathered from discussions and reading, is that the appropriate equivalence relation is diffeomorphism. However, despite this common understanding, I've been unable to find a proof (or even a formal statement) establishing diffeomorphism as the correct equivalence in any authoritative source.
Statement A. Any closed, oriented, PL 4-manifold admits a unique compatible smooth structure (up to diffeomorphism).
Question. Is Statement A correct?
Please provide a formal proof or a direct reference to a peer-reviewed paper or authoritative source establishing its validity. The cited sources [1], [2], [3], [4], and [5] were found insufficient for this purpose. For example, despite [1] has been often recommended as the reliable source, see footnote [a] below; [3] serves as a nice and modern survey, but it addresses more on TOP-DIFF, and says little about PL-DIFF (cf footnote [c] below).
Common (Yet Failing) Intuitive Proof Sketch
In the remainder of this post, I will proceed by outlining a proof sketch for Statement A, acknowledging its common appeal, and subsequently explaining its inadequacy.
Given an oriented, closed, PL 4-manifold, there are at least three equivalence relations on the set of its compatible smooth structures: concordance, isotopy, and diffeomorphism [2, p.11]. The distinction is crucial.
We note the implications: concordance ⇐ isotopy ⇒ diffeomorphism [2, p.11]. Diffeomorphism is the primary focus here; its classification is achieved using Kirby diagrams and moves [4]. Concordance often allows for deeper understanding via algebraic-homotopy/obstruction theory techniques, at least in higher dimensions ($d \geq 5, 6$) [2, Preface].
A cornerstone result, the "Concordance Implies Isotopy Theorem," establishes the equivalence concordance ⇔ isotopy [1][2, p.11]. This result, combined with the definitions, yields the full set of implications (diffeomorphism's failure to be equivalent is noted on the same page): concordance ⇔ isotopy ⇒ diffeomorphism
An intuitive proof sketch for Statement A would proceed as follows:
Prove a modified Statement A is correct up to concordance (instead of diffeomorphism). This should be achieved using algebraic-homotopy techniques (e.g., demonstrating the contractibility of a relevant space (perhaps $PL(4)/O(4)$?)).
Conclude that, by the Concordance Implies Isotopy Theorem (C⇒I), and the relation I⇒D, Statement A is correct up to diffeomorphism.
Potential Failure of Step 1: I suspect the obstruction theoretic technique may not be applicable to the 4-dimensional Concordance Problem. The literature, including references [1], [2], and [3], frequently indicates that this technique is most effective, and often restricted, to dimensions $\geq 5$ (cf footnote [b]). Consequently, even if a relevant classifying space (perhaps $PL(4)/O(4)$?) were proven to be contractible, this fact would likely not yield a proof for the modified Statement A.
Potential Failure of Step 2: The Concordance implies Isotopy Theorem also appears to be subject to a similar dimensional constraint, typically holding only for $m \geq 5$. This lack of explicit dimensional clarity in foundational texts, particularly the highly and frequently recommended [1], is a source of frustration. The necessary dimensional assumptions often seem to be embedded implicitly within the context of the overall theory, leading to ambiguity regarding the theorem's applicability in the critical dimension of 4 (see footnote [a]).
Footnote
[a] In [1, Essay V, Section 0], it clearly states,
We have largely neglated the classical problem of classification of DIFF structures on M compatible with a fixed PL structure. Fortunately, C. Rourke [Ro2] has announced an exposition emphasizing this problem.
In [1], the reference [Ro2] was the 1972 preprint 'On structure theorem', which does not seem to exist anymore now (half century later).
[b] In [3, 4.5 Notes, Note: Topological manifolds and smoothings, p.207], it states,
we will outline the theory of topological manifolds and of their smooth structures. The theory works best in dimension 5 or more, where it offers complete answers on the existence and classification of smooth structures on topological manifolds. The theory is quite weaker in dimension 4, but it is still relevant.
[c] In [3, 4.5 Notes, Note: Topological manifolds and smoothings, p.219], it acknowledges that they do not adddress PL-SMOOTH well, pointing to the readers some relevant references. Those references are essentially [2] (which focuses on the homotopic tools on the concordance problem in $d \geq 5$) and [5] (which focuses on establishing the existence of smoothings as an obstruction problem).
Reference
- [1] Foundational Essays on Topological Manifolds, Smoothings, and Triangulations-[R Kirby, L Siebenmann]
- [2] Smoothings of Piecewise Linear Manifolds-[M Hirsch, B Mazur]
- [3] The Wild World of 4-Manifolds-[A Scorpan]
- [4] 4-manifolds and Kirby calculus-[A Stipsicz, R Gompf]
- [5] Obstructions to the Smoothing of Piecewise-Differentiable Homeomorphisms-[James Munkres]
