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In this work, we design a nearly linear time discrete Morse theory basedalgorithm for computing homology groups of 2-manifolds, thereby establishingthe fact that computing homology groups of 2-manifolds is remarkably easy.Unlike previous algorithms of similar flavor, our method works withcoefficients from arbitrary abelian groups. Another advantage of our methodlies in the fact that our algorithm actually elucidates the topological reasonthat makes computation on 2-manifolds easy. This is made possible owing to anew simple homotopy based construct that is referred to as \emph{expansionframes}. To being with we obtain an optimal discrete gradient vector fieldusing expansion frames. This is followed by a pseudo-linear time dynamicprogramming based computation of discrete Morse boundary operator. Theefficient design of optimal gradient vector field followed by fast computationof boundary operator affords us near linearity in computation of homologygroups. Moreover, we define a new criterion for nearly optimal Morse functionscalled pseudo-optimality. A Morse function is pseudo-optimal if we can obtainan optimal Morse function from it, simply by means of critical cellcancellations. Using expansion frames, we establish the surprising fact that anarbitrary discrete Morse function on 2-manifolds is pseudo-optimal.

Optimal Morse functions and $H(\mathcal{M}^2,\mathbb{A})$ in $\tilde{O}(N)$ time