This paper studies the space of r-tuples of 2 x 2 complex matrices that generate the full matrix algebra, considered up to change-of-basis. We show that when r is 2, this space is homotopy equivalent to the quotient of a product of a circle and a sphere by an involution. When r is greater than 2, we determine the rational cohomology of the space in degrees less than 4r-6. As an application, we use the machinery of [5] to prove that for all natural numbers d, there exists a ring R of Krull dimension d and a degree-2 Azumaya algebra A over R that cannot be generated by fewer than 2[ d/4 ] + 2 elements.

We acknowledge the support of the Natural Sciences and Engineering Research Council of Canada (NSERC), RGPIN-2016-03780, RGPIN-2021-02603. Cette recherche a ete financee par le Conseil de recherches en sciences naturelles et en genie du Canada (CRSNG), RGPIN-2016-03780, RGPIN-2021-02603.