**abstract:**
Combinatorial vector bundles are discrete analogues of real vector bundles introduced by Robert
MacPherson in 1993. He built on joint work with Gelfand which can be reinterpreted as stating a connection between
the rational cohomology of the classifying space of real vector bundles -- the Grassmannian -- and that of the
classifying space of combinatorial vector bundles -- the combinatorial Grassmannian, nowadays also called the
MacPhersonian. This latter space is the order complex of the poset of all oriented matroids of a fixed rank, and is
conjectured to be homotopy equivalent to the Grassmannian: this is the MacPhersonian conjecture.
We use MacPherson's discretization map to write the real Grassmannian as the colimit of a diagram of spaces over
the corresponding MacPhersonian, enabling us to understand the topology of the combinatorial Grassmannian by
studying a new class of subspaces of the real Grassmannian. Applying homotopy-theoretical methods, in particular
homotopy colimits, it turns out these spaces know a lot about the conjecture: for example, if all of them are
contractible then the conjecture holds, and this lets us reprove all previously known cases in a uniform way. On the
other hand, with the help of this diagram we establish several spectral sequences relating the (co)homology of the
Grassmannian and of the combinatorial Grassmannian. I present how our diagram can be constructed, and how
topological information regarding the MacPhersonian can be extracted from it.
The talk is based on joint work with Pavle Blagojević.

Thu 3 Oct, 17:20 - 18:00, Aula Dini

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