Homotopy Classification of Projections in the Corona Algebra of a Non-simple C-algebra

We study projections in the corona algebra of $C\left( X \right)\,\otimes \,K$ , where $K$ is the ${{C}^{*}}$ -algebra of compact operators on a separable infinite dimensional Hilbert space and $X\,=\,[0,\,1],\,[0,\,\infty ),\,(-\infty ,\,\infty ),\,\text{or}\,\text{ }\!\![\!\!\text{ 0,}\,\text{1 }\!\!]\!\!\text{ / }\!\!\{\!\!\text{ 0,}\,\text{1 }\!\!\}\!\!\text{ }$ . Using BDF's essential codimension, we determine conditions for a projection in the corona algebra to be liftable to a projection in the multiplier algebra. We also determine the conditions for two projections to be equal in ${{K}_{0}}$ , Murray-von Neumann equivalent, unitarily equivalent, or homotopic. In light of these characterizations, we construct examples showing that the equivalence notions above are all distinct.