| Register
\(\newcommand{\Cat}{{\rm Cat}} \) \(\newcommand{\A}{\mathcal A} \) \(\newcommand{\freestar}{ \framebox[7pt]{$\star$} }\)

1. Trisections of 4-manifolds

Let $X$ be a closed, oriented and smooth $4$-manifold.


    1. Add remark

      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.


        • Add remark

          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 $i=1,2,3$.


            • Add remark

              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.)


                • Add remark

                  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?


                    • Add remark

                      Problem 1.1.

                      Is the trisection genus additive?
                          (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.)


                        • Add remark

                          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?


                            • Add remark

                              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?


                                • Add remark

                                  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?


                                    • Add remark

                                      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?


                                        • Add remark

                                          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?


                                            • Add remark

                                              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 $S^4$?


                                                • Add remark

                                                  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?


                                                    • Add remark

                                                      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.


                                                        • Add remark

                                                          Problem 1.28.

                                                          Find new invariants of smooth $4$-manifolds using Heegaard Floer homology techniques.


                                                            • Add remark

                                                              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.

                                                              In particualr, which one of following complexes give more information:
                                                              1. (a) Pants complex
                                                              2. (b) Curve complex
                                                              3. (c) Cut system


                                                                • Add remark

                                                                  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?


                                                                    • Add remark

                                                                      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?


                                                                        • Add remark

                                                                          Problem 1.36.

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


                                                                            • Add remark

                                                                              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 $(X,F^2)$.


                                                                                • Add remark

                                                                                  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?


                                                                                    • Add remark

                                                                                      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.