Spaces of generators for the 2 x 2 complex matrix algebra

This paper studies the space of r-tuples of 2x2 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 2left perpendiculard/4right perpendicular+ 2 elements.