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The genus of a closed orientable 3-manifold $M^3$ is the minimum genus among all Heegaard splitting surfaces for $M$. Every such 3-manifold bounds a compact 4-manifold. Let $I(M)$ denote the minimum second Betti number $b_2$ amongst all such bounding $X^4$.

Is there a sequence of genus 2 3-manifolds $M_n$ such that $I(M_n) \to \infty$.

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  • $\begingroup$ Do you know any sequence of $M_n$ such that $I(M_n)--> \infty $? $\endgroup$
    – Anubhav Mukherjee
    Commented Nov 19, 2019 at 16:14
  • $\begingroup$ @Mukherjee I do not but I have a faint memory of seeing a lower bound for $I(M)$ somewhere. It probably was not called $I(M)$ though. $\endgroup$
    – user101010
    Commented Nov 20, 2019 at 12:31
  • $\begingroup$ It will be great if you can recall a source for that. I am not able to think of such examples. $\endgroup$
    – Anubhav Mukherjee
    Commented Nov 20, 2019 at 14:47
  • $\begingroup$ I might be wrong, but I think one can get a Kirby diagram for a bounding manifold consisting of genus component $0$-framed unlink and multiple $\pm 1$ framed unknots unlinked from each other. Then one can eliminate the $\pm 1$ framed unknots by a Kirby move. So the answer is no. There is a description of this diagram, for example, in "A simple proof of the fundamental theorem of Kirby calculus on links" by Ning Lu. $\endgroup$
    – mathquest
    Commented Nov 21, 2019 at 1:35
  • $\begingroup$ @mathquest I didn;t get your comment. What are you trying to prove/state here? $\endgroup$
    – Anubhav Mukherjee
    Commented Nov 21, 2019 at 14:27

1 Answer 1

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I think it follows that a Heegaard genus 2 manifold will bound a topological 4-manifold with $b_2\leq 2$ from a theorem of Steve Boyer.

Boyer, Steven, Simply-connected 4-manifolds with a given boundary, Trans. Am. Math. Soc. 298, 331-357 (1986). ZBL0615.57008. MR0857447

I'm guessing that you're interested in the smooth category though?

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  • $\begingroup$ If we don't restrict it to just genus 2, do you know any sequence M_n, such that this limit will be infinity, i.e my first comment in the comment section. (In smooth category). $\endgroup$
    – Anubhav Mukherjee
    Commented Nov 24, 2019 at 6:01
  • $\begingroup$ Infact starting with any manifold, we can consider one of it's open book with connected binding. Then by doing a zero surgery,i.e, adding a 0 framed 2 handle in 4 dim version we can construct a surface bundle over circle. Now the question is, can we say anything about such special class? If it is trivial bundle, then we can conclude something. $\endgroup$
    – Anubhav Mukherjee
    Commented Nov 24, 2019 at 6:06
  • $\begingroup$ I'm not sure - one can get lower bounds on bounding simply-connected manifolds in terms of $rank(H_1(M))$, but I don't think this gives a lower bound on $b_2$ of non-simply connected bounding manifolds. $\endgroup$
    – Ian Agol
    Commented Nov 24, 2019 at 7:36
  • $\begingroup$ @IanAgol: indeed, it doesn't give a lower bound in the non-simply connected case; e.g. just take a boundary connected sum of N copies of any rational homology 4-ball (like $T^*\mathbb{RP}^2$). $\endgroup$
    – Marco Golla
    Commented Nov 24, 2019 at 9:31
  • $\begingroup$ @IanAgol Yeah I was primarily interested in the smooth case - but I'm also happy to hear about the topological case. Which theorem in Boyer's paper were you looking at? $\endgroup$
    – user101010
    Commented Nov 26, 2019 at 13:47

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