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Whilst reading 'the book' I stumbled across the following elegant little theorem, due to Gómez-Larrañaga and González-Acuña.

Let $M$ be a closed $3$-dimensional manifold. Then its Lusternik-Schnirelmann category satisifes

$$cat(M)=\begin{cases}1&\text{if}\;\;\pi_1M=\{1\}\\ 2&\text{if}\;\;\pi_1M\;\;\text{is free}\\ 3&\text{otherwise}.\end{cases}$$

I thought this was a pretty neat little characterisation. My question, then, is if anyone has ever sat down and tried to produce a similar statement for 4-manifolds? If possible I would be looking for a pleasingly tenuous connection to some of the smooth invariants that people like to write down.

I'm aware of a few scattered results in the area. For example Dranishnikov, Katz and Rudyak have a joint paper in which they formulate some general statements for higher dimensional manifolds. The book itself also includes a discussion of Cornea, Lupton, Oprea, Tanre and other's work on the relation of the categorical invariants to symplectic topology in general, but again they are not focused necessarily on the 4-dimensional case.

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    $\begingroup$ One observation is that since category is homotopy invariant, it wouldn't make much sense to try to connect it to invariants of the smooth structure. Another observation is that category of $4$-manifolds does not admit a similar description just in terms of the fundamental group (since e.g., $cat(S^4)=1$ and $cat(\mathbb{C}P^2)=2$). $\endgroup$
    – Mark Grant
    Commented Jun 26, 2019 at 13:07
  • $\begingroup$ @MarkGrant, thanks for the interest. I certainly wouldn't be too hopeful of any real link with the smooth invariants. My question in that direction actually stems from the contention that much of the purely topological information in them is not well filtered out, and they are often not quite as strong as some may want (Here I'm probably thinking more in terms of homology of diffeomorphism and embedding spaces, for instance, than Donaldson Invariants). As for a 4-dimensional cat statement, the obvious idea would be that the intersection form (at least) would appear in there, alongside $\pi_1$. $\endgroup$
    – Tyrone
    Commented Jun 26, 2019 at 14:04
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    $\begingroup$ Is it known that all closed simply connected 4-manifolds $X$ have $cat(X) =1$ or $cat(X) =2$? I know that it is a conjecture that all of the smooth ones can be described with only 0,2, and 4 handles. $\endgroup$
    – user101010
    Commented Jun 26, 2019 at 18:13
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    $\begingroup$ @user101010: Yes, that's right. For an $(r-1)$-connected manifold $X$ one has $cat(X)\leq dim(X)/r$. In fact, by Poincare duality a simply connected $4$-manifold is either $S^4$ or has category $2$. $\endgroup$
    – Mark Grant
    Commented Jun 26, 2019 at 19:23
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    $\begingroup$ A $4$-dimensional compact manifold with free $\pi_1$ has L-S category $2$. $\endgroup$
    – Jeff Strom
    Commented Jun 27, 2019 at 23:31

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