Kazunori KIKUCHI

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Differential topology
Keywords Four-dimensional, manifold, differential structure

I have been studying topology of smooth four-dimensional manifolds, in particular interested in homology genera, representations of diffeomorphism groups to intersection forms, and branched coverings. Let me give a simple explanation of what interests me the most, or homology genera. The homology genus of a smooth four-dimensional manifold M is a map associating to each two-dimensional integral homology class [x] of M the minimal genus g of smooth surfaces in M that represent [x]. For simplicity, reducing the dimensions of M and [x] to the halves of them respectively, consider as a two-dimensional manifold the surface of a doughnut, or torus T, and a one-dimensional integral homology class [y] of T. Draw a meridian and a longitude on T as on the terrestrial sphere, and let [m] and [l] denote the homology classes of T represented by the meridian and the longitude respectively. It turns out that [y] = a[m] + b[l] for some integers a and b, and that [y] is represented by a circle immersed on T with only double points. Naturally interesting then is the following question: what is the minimal number n of the double points of such immersions representing [y]? Easy experiments would tell you that, for example, n = 0 when (a,b) = (1,0) or (0,1) and n = 1 when (a,b) = (2,0) or (0,2). In fact, it is proved with topological methods that n = d−1, where d is the greatest common divisor of a and b. It is the minimal number n for T and [y] that corresponds to the minimal genus g for M and [x]. The study on the minimal genus g does not seem to proceed with only topological methods; it sometimes requires methods from differential geometry, in particular methods with gauge theory from physics; though more difficult, it is more interesting to me. I have been tackling the problem on the minimal genus g with such a topological way of thinking as to see things as if they were visible even though invisible.