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## 1. Trisections of 4-manifolds

Let $X$ be a closed, oriented and smooth $4$-manifold.
1. #### Problem 1.02.

Let $\Sigma$ be an embedded surface in $X$ which bounds a handle body. Assume that the map $\pi_1(\Sigma)\rightarrow \pi_1(X)$ induced by the inclusion of $\Sigma$ in $X$ is surjective and $2+g(\Sigma)\ge\chi(X)$. Is there a trisection of $X$ whose core is $\Sigma$? If not, what other conditions we should add to get such a trisection.
• #### Problem 1.04.

Suppose $T_1$ and $T_2$ are balanced trisections of $X$, with the same core $\Sigma\subset X$. Is there an ambient isotopy $\phi:X\rightarrow X$ which maps $T_1$ to $T_2$ i.e. $\phi(X_i^1)=X_i^2$ for any $i=1,2,3$.
• #### Problem 1.06.

A trisection of $S^4$ is called standard if it is a stabilization of the genus zero trisection. Is there a non-standard balanced or unbalanced trisection of $S^4$?
(Claim: A trisection is diffeomorphic to a standard trisection iff it is isotopic to a standard trisection.)
• #### Problem 1.08.

A balanced trisection is called minimal if its core has the smallest possible genus. Does there exist a non-minimal balanced trisection of $X$ which is not a stabilization of a minimal trisection?
• #### Problem 1.1.

(Note: If yes, then it implies the Poincare conjecture by connected sum. Assume $X$ is a fake $S^4$. Then for $k$ sufficiently large $X\#_kS^2\times S^2$ is diffeomorphic to $\#_kS^2\times S^2$. Thus trisection genus of $X$ is zero.)
• #### Problem 1.12.

Is the unbalanced trisection genus of $X$ equal to the balanced trisection genus? Or equivalently, is there an example where unbalanced genus is strictly smaller than balanced genus?
• #### Problem 1.14.

Develop a notion of equivariant trisection; given an action of a finite group on $X$ which preserves the trisection in an appropriate sense, what does it say about the trisection?
• #### Problem 1.16.

Given a smooth map (if necessary assume it is homotopy equivalence and degree one) $f:X\rightarrow Y$, does there exist trisections of $X$ and $Y$ such that $f$ is homotopic to a map that preserves "strata" of the trisections?
• #### Problem 1.18.

Assume $X$ is the double of a smooth $4$-manifold with boundry $Y$. What can we say about the trisections of $X$? For example suppose $Y$ is obtained from surgery on an embedded link $L$ in $\#_kS^1\times S^2$. Is the induced trisection on $X$ reducible?
• #### Problem 1.2.

A Morse $2$-function to a disk can be homotoped to a trisected Morse $2$-function. What about Morse $2$-functions to higher genus surfaces with boundary?
• #### Problem 1.22.

Suppose $X$ is a homotopy $S^4$ together with a trisection such that $X_1$ is diffeomorphic to $B^4$. Is $X$ diffeomorphic to $S^4$?
• #### Problem 1.24.

What Reimannian geometry conditions imply about trisections e.g. Assume $X$ is hyperbolic with large injectivity radius, does that imply the trisection genus of $X$ is large?
• #### Problem 1.26.

Translate $4$-manifold operations e.g. rational blow downs, Gluck twist, corks,... to trisections. Find trisection diagrams for exotic pairs of $4$-manifolds.
• #### Problem 1.28.

Find new invariants of smooth $4$-manifolds using Heegaard Floer homology techniques.
• #### Problem 1.3.

Find an algorithm to translate a Heegaard diagram of $\#_k S^1\times S^2$ to a standard one via isotopies and handleslides. Find an invariant which gives a bound on the number of handleslides.

1. (a) Pants complex
2. (b) Curve complex
3. (c) Cut system
• #### Problem 1.32.

What invariants of $4$-manifolds can be computed using the trisection diagrams (i.e. use combinatorial information of curve intersections) What invariants are not computable?

Give two trisections of a $4$-manifold, can you get a bound for the number of required stabilizations to make them diagrams for the same trisection?
• #### Problem 1.34.

Given a Lefschetz fibration of $X$ over a disk. Can you convert it to a trisection? Characterize which trisections arise from this construction.

What about Lefschetz fibrations over a closed surface? Can you make an irreducible trisection?
• #### Problem 1.36.

Develop a notion of trisection for higher dimensional manifolds. What is the trisection for exotic $7$-spheres?
• #### Problem 1.38.

Study Spin bordism group. Find a trisection interpretation for a pair $(X,F^2)$ where $[F^2]=w_2(X)$ or general bordiam $(X,F^2)$.
• #### Problem 1.4.

A $(g,g)$-balanced trisection is reducible and it is a diagram for a connected sum of copies of $S^1\times S^3$’s. If for a $(g,k)$-trisection we have that $\frac{k}{g}\approx 1$, is it reducible? Is every $(g,0)$-trisection irreducible?
• #### Problem 1.42.

Let $X$ be the double of the non-trivial disk bundle over $\mathbb{RP}^2$. Is $l(X)=7$? Find a closed smooth $4$-manifold $X$ such that $l(X)$ is between $0$ and $7$.

Cite this as: AimPL: Trisections and low-dimensional topology, available at http://aimpl.org/trisections.