There are 160,839<1> + 160,650<-1> 3-planes in a 7-dimensional cubic hypersurface

Abstract: The expression in the title is a bilinear form and it comes from an Euler number in A1-algebraic topology. Such Euler numbers can be constructed with Hochschild homology, self-duality of Koszul complexes, pushforwards in SL_c oriented cohomology theories, and sums of local degrees. We show an integrality result for A1-Euler numbers and apply this to the enumeration of d-planes in complete intersections. Classically such counts are valid over C and sometimes extended to the real numbers, but A1-homotopy theory allows one to perform counts over a large class of fields, and records information about the solutions in bilinear form. The example in the title then follows from work of Finashin--Kharlamov. This is joint work with Tom Bachmann.

algebraic topology

Audience: researchers in the topic

Online algebraic topology seminar

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