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I found the notion "guts of three-manifolds" unclear to me. There exists "sutured guts" and "pared guts" in the literature, the well definedness of both are vague to me.

Sutured guts: The following definition of sutured guts is cited from the preprint Guts in Sutured Decompositions and the Thurston Norm by Agol and Zhang. Let $(M,\gamma)$ be a taut sutured 3-manifold, where $\gamma$ is a collection of disjoint annuli and tori on the boundary $\partial M$. Let $R(\gamma)=\partial M\setminus int(\gamma)=R_+\cup R_-$, where $R_+$ and $R_-$ are oriented according to the orientation of $\gamma$. We further define

  • A product disk is a disk $D$ in $M$ such that $D\cap \gamma$ consists of two arcs.
  • A product annulus is an essential annulus $A$ in $M$ such that one component of $A$ lies on $R_-$ and another component of $R_+$ lies on $R_-$.

Decompose $(M,\gamma)$ along a maximal collection of disjoint nonparallel product annuli and product disks, throw away all product components (called the window) of the resulting sutured manifold, the remaining sutured manifold $(\overline M,\overline \gamma)$ is called the sutured guts of $(M,\gamma)$. It is well defined up to isotopy by JSJ-decomposition.

Pared guts: In this artical The minimal volume orientable hyperbolic 2-cusped 3-manifold by Agol, the guts of a pared manifold $(M,P)$ is defined as follows. A pared manifold $(M,P)$ is a pair where $M$ is a compact orientable irreducible 3-manifold and $P$ is a union of essential annuli and tori on the boundary $\partial M$, such that every abelian noncyclic subgroup of $\pi_1(M)$ is peripheral with respect to $P$, and every mapping of an annulus $(A,\partial A)\rightarrow (M,P)$ that is injective on the fundamental group can be homotoped into $P$.

Let $(M,P)$ be a pared manifold such that $\partial_0M:=\partial M\setminus P$ is incompressible. There is a canonical family of annuli (I suspect that one should also consider tori here?) $(A,\partial A)\subset (M,\partial_0M)$ which is the maximal collection of nonparallel essential annuli such that every other essential annulus $(B,\partial B)\subset (M,\partial_0M)$ can be relatively isotoped to be disjoint with $A$. A components of $M\setminus A$ is called a window if it is an ($I$-bundle, $\partial I$-bundle). Let $W$ be the union of such components. Then $M\setminus W$ is called the pared guts of $(M,P)$, which is also a pared manifold, with pared locus $\partial(M\setminus W)-\partial_0M$.

My question is:

  • How to apply the JSJ-decomposition to see the well definedness of the sutured guts? I thought about the Characteristic Pair Theorem to produce a canonical seifert pair $(\Sigma,\Phi)\subset (M,R(\gamma))$, the frontier of the $I$-bundle components of $(\Sigma, \Phi)\subset (M,R(\gamma))$ seems to be the desired maximal collection of product annuli and product disks, but I am not sure. I also found that in the paper The virtual fibering theorem for 3-manifolds by Friedl and Kitayama, before Lemma 3.2 they said the guts may depend on the choice of the disks and annuli, which confuses me.
  • Why is the pared guts well defined? In other words, why is there a canonical set of annuli as in the definition? Is it from the Characteristic Pair Theorem?
  • What is the relation between pared guts and the sutured guts? Why do we need to consider product disks in the sutured case, but not in the pared case?
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Edited: to reflect the correct definitions.


Question 1: Why are the guts well-defined?

Answer 1: By the JSJ theory there is a unique collection of $I$-bundles (and Seifert fibered spaces) that contain (up to isotopy) all essential product disks and product annuli (and all essential tori). We throw away the $I$-bundles. Note that any Seifert fibered pieces remaining are pared solid tori. (There are no interesting Seifert fibered pieces because the original manifold $M$ has "non-degenerate" Thurston norm. So all essential tori in $M$ are parallel to boundary components.)

Question 2: Why is the pared guts well defined?

Answer 2: For exactly the same reason.

Question 3: What is the relation between pared guts and the sutured guts?

Answer 3: Pared guts are directed at understanding geometry. Sutured guts have an additional homological condition (the assumption of tautness).

Question 4: Why do we need to consider product disks in the sutured case, but not in the pared case?

Answer 4: We don't need them in either case. Consider the frontier of a regular neighbourhood of the union of a product disk and the sutures it meets.

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  • $\begingroup$ I have more questions to Answer 1: the characteristic pair theorem gives Seifert submanifolds and I-bundles. The Seifert submanifolds contain annuli as well. should I further decompose the Seifert submanifolds along annuli to produce I-bundles and solid tori with sutures? Is there any reference for this argument? $\endgroup$
    – Fredy
    Feb 20 at 10:37
  • $\begingroup$ We do not decompose the Seifert fibered spaces. Instead we throw them away. $\endgroup$
    – Sam Nead
    Feb 21 at 8:38
  • $\begingroup$ For a general reference on sutured manifold theory, I recommend Martin Scharlemann’s articles “Lectures on the theory of sutured 3-manifolds” and (the much longer) “Sutured manifolds and generalised Thurston norms”. $\endgroup$
    – Sam Nead
    Feb 21 at 8:41
  • $\begingroup$ Why not decompose the Seifert manifolds? They may contain essential annuli as well, I thought we had to cut along those annuli to result in an I-bundle (which we throw away) and some solid torus with sutures on the boundary (which we keep as guts), for example, let M be a Seifert manifold with two sutures on the boundary, such that the two sutures are parallel to the fiber of M, it contains non-trivial guts, right? $\endgroup$
    – Fredy
    Feb 21 at 9:25
  • $\begingroup$ I've edited my answer to more directly address the (lack of) Seifert fibered pieces. Previously I was not using the hypotheses correctly. $\endgroup$
    – Sam Nead
    Feb 21 at 11:18

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