## Tuesday Seminar on Topology

Seminar information archive ～09/18｜Next seminar｜Future seminars 09/19～

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

### 2015/06/23

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

Box complexes and model structures on the category of graphs (JAPANESE)

**Takahiro Matsushita**(The University of Tokyo)Box complexes and model structures on the category of graphs (JAPANESE)

[ Abstract ]

To determine the chromatic numbers of graphs, so-called the graph

coloring problem, is one of the most classical problems in graph theory.

Box complex is a Z_2-space associated to a graph, and it is known that

its equivariant homotopy invariant is related to the chromatic number.

Csorba showed that for each finite Z_2-CW-complex X, there is a graph

whose box complex is Z_2-homotopy equivalent to X. From this result, I

expect that the usual model category of Z_2-topological spaces is

Quillen equivalent to a certain model structure on the category of

graphs, whose weak equivalences are graph homomorphisms inducing Z_2-

homotopy equivalences between their box complexes.

In this talk, we introduce model structures on the category of graphs

whose weak equivalences are described as above. We also compare our

model categories of graphs with the category of Z_2-topological spaces.

To determine the chromatic numbers of graphs, so-called the graph

coloring problem, is one of the most classical problems in graph theory.

Box complex is a Z_2-space associated to a graph, and it is known that

its equivariant homotopy invariant is related to the chromatic number.

Csorba showed that for each finite Z_2-CW-complex X, there is a graph

whose box complex is Z_2-homotopy equivalent to X. From this result, I

expect that the usual model category of Z_2-topological spaces is

Quillen equivalent to a certain model structure on the category of

graphs, whose weak equivalences are graph homomorphisms inducing Z_2-

homotopy equivalences between their box complexes.

In this talk, we introduce model structures on the category of graphs

whose weak equivalences are described as above. We also compare our

model categories of graphs with the category of Z_2-topological spaces.