Homotopy type of shellable q-complexes and their homology groups

The theory of shellable simplicial complexes brings together combinatorics, algebra, and topology in a remarkable way. Initially introduced by Alder for q-simplicial complexes, recent work of Ghorpade, Pratihar, and Randrianarisoa extends the study of shellability to q-matroid complexes and determines singular homology groups for a subclass of these q-simplicial complexes. In this paper, we determine the homotopy type of shellable q-simplicial complexes. Moreover, we establish the shellability of order complexes from lexicographically shellable q-simplicial complexes, that include the q-matroid complexes. This results in a comprehensive determination of the homology groups for any lexicographically shellable q-complexes.