I've recently started stydying algebraic topology and am now learning about cell complexes. I understand the iterative construction of such spaces, but I lack any intuition concerning the limitations on the possible resulting structures.
I thus have two questions for which I am seeking, preferably, non-highly technical answers:
1) What types of structures can, and cannot, be created iteratively from cells?
2) What types of spaces are, and are not, homotopic to the spaces created from cells?