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   Manolescu, Ciprian (January 2018, Proceedings of the International Congress of Mathematicians)

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   The study of triangulations on manifolds is closely related to understanding the three-dimensional homology cobordism group. We review here what is known about this group, with an emphasis on the local equivalence methods coming from Pin.2/- equivariant Seiberg-Witten Floer spectra and involutive Heegaard Floer homology. 

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   For integers $$n\ge 0$$, an iterated triangulation $$\mathrm{Tr}(n)$$ is defined recursively as follows: $$\mathrm{Tr}(0)$$ is the plane triangulation on three vertices and, for $$n\ge 1$$, $$\mathrm{Tr}(n)$$ is the plane triangulation obtained from the plane triangulation $$\mathrm{Tr}(n-1)$$ by, for each inner face $$F$$ of $$\mathrm{Tr}(n-1)$$, adding inside $$F$$ a new vertex and three edges joining this new vertex to the three vertices incident with $$F$$. In this paper, we show that there exists a 2-edge-coloring of $$\mathrm{Tr}(n)$$ such that $$\mathrm{Tr}(n)$$ contains no monochromatic copy of the cycle $$C_k$$ for any $$k\ge 5$$. As a consequence, the answer to one of two questions asked by Axenovich et al. is negative. We also determine the radius 2 graphs $$H$$ for which there exists $$n$$ such that every 2-edge-coloring of $$\mathrm{Tr}(n)$$ contains a monochromatic copy of $$H$$, extending a result of Axenovich et al. for radius 2 trees. 

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