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Given two discrete Morse functions on a simplicial complex, we introduce the connectedness homomorphism between the corresponding discrete Morse complexes. This concept leads to a novel framework for studying the connectedness in discrete Morse theory at the chain complex level. In particular, we apply it to describe a discrete analogy to 'cusp-degeneration' of Morse complexes. A precise comparison between smooth case and our discrete cases is also given.
Chong Zheng (Wed,) studied this question.