## Tuesday Seminar on Topology

Seminar information archive ～03/27｜Next seminar｜Future seminars 03/28～

Date, time & place | Tuesday 17:00 - 18:30 Room #056 (Graduate School of Math. Sci. Bldg.) |
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Organizer(s) | KOHNO Toshitake, KAWAZUMI Nariya, KITAYAMA Takahiro, SAKASAI Takuya |

Remarks | Tea: 16:30 - 17:00 Common Room |

**Future seminars**

### 2020/04/07

17:00-18:30 Room #056 (Graduate School of Math. Sci. Bldg.)

Gauge theory and the diffeomorphism and homeomorphism groups of 4-manifolds (JAPANESE)

**Hokuto Konno**(RIKEN iTHEMS)Gauge theory and the diffeomorphism and homeomorphism groups of 4-manifolds (JAPANESE)

[ Abstract ]

I will explain my recent collaboration with several groups that develops gauge theory for families

to extract difference between the diffeomorphism groups and the homeomorphism groups of 4-manifolds.

After Donaldson’s celebrated diagonalization theorem, gauge theory has given strong constraints on the topology of smooth 4-manifolds. Combining such constraints with Freedman’s theory, one may find many non-smoothable topological 4-manifolds.

Recently, a family version of this argument was started by T. Kato, N. Nakamura and myself, and soon later it was developed also by D. Baraglia and his collaborating work with myself. More precisely, considering gauge theory for smooth fiber bundles of 4-manifolds, they obtained some constraints on the topology of smooth 4-manifold bundles. Using such constraints, they detected non-smoothable topological fiber bundles of smooth 4-manifolds. The existence of such bundles implies that there is homotopical difference between the diffeomorphism and homeomorphism groups of the 4-manifolds given as the fibers.

If time permits, I will also mention my collaboration with Baraglia which shows that a K3 surface gives a counterexample to the Nielsen realization problem in dimension 4. This example reveals also that there is difference between the Nielsen realization problems asked in the smooth category and the topological category.

I will explain my recent collaboration with several groups that develops gauge theory for families

to extract difference between the diffeomorphism groups and the homeomorphism groups of 4-manifolds.

After Donaldson’s celebrated diagonalization theorem, gauge theory has given strong constraints on the topology of smooth 4-manifolds. Combining such constraints with Freedman’s theory, one may find many non-smoothable topological 4-manifolds.

Recently, a family version of this argument was started by T. Kato, N. Nakamura and myself, and soon later it was developed also by D. Baraglia and his collaborating work with myself. More precisely, considering gauge theory for smooth fiber bundles of 4-manifolds, they obtained some constraints on the topology of smooth 4-manifold bundles. Using such constraints, they detected non-smoothable topological fiber bundles of smooth 4-manifolds. The existence of such bundles implies that there is homotopical difference between the diffeomorphism and homeomorphism groups of the 4-manifolds given as the fibers.

If time permits, I will also mention my collaboration with Baraglia which shows that a K3 surface gives a counterexample to the Nielsen realization problem in dimension 4. This example reveals also that there is difference between the Nielsen realization problems asked in the smooth category and the topological category.

### 2020/04/14

17:00-18:30 Room #056 (Graduate School of Math. Sci. Bldg.)

Homology of right-angled Artin kernels (ENGLISH)

**Daniel Matei**(IMAR Bucharest)Homology of right-angled Artin kernels (ENGLISH)

[ Abstract ]

The right-angled Artin groups $A(G)$ are the finitely presented groups associated to a finite simplicial graph $G=(V,E)$, which are generated by the vertices $V$ satisfying commutator relations $vw=wv$ for every edge $vw$ in $E$. An Artin kernel $N_h(G)$ is defined by an epimorphism $h$ of $A(G )$ onto the integers. In this talk, we discuss the module structure over the Laurent polynomial ring of the homology groups of $N_h(G)$.

The right-angled Artin groups $A(G)$ are the finitely presented groups associated to a finite simplicial graph $G=(V,E)$, which are generated by the vertices $V$ satisfying commutator relations $vw=wv$ for every edge $vw$ in $E$. An Artin kernel $N_h(G)$ is defined by an epimorphism $h$ of $A(G )$ onto the integers. In this talk, we discuss the module structure over the Laurent polynomial ring of the homology groups of $N_h(G)$.