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    Title   l  Orbifold Morse–Smale–Witten complexes
    Journal   l  International Journal of Mathematics
    Year   l  2014
    Vol   l  25
    No   l  5
    Pages   l  Doi:10.1142
    Author   l  Cheol-Hyun Cho
    DownLoad    l 1398044777_0.034752.pdf  
Given aMorse–Smale function on an effective orientable orbifold, we construct its Morse–
Smale–Witten complex. We show that critical points of a certain type have to be discarded
to build a complex properly, and that gradient flows should be counted with
suitable weights. Its homology is proven to be isomorphic to the singular homology of
the quotient space under the self-indexing assumption. For a global quotient orbifold
[M/G], such a complex can be understood as the G-invariant part of the Morse complex
of M, where the G-action on generators of the Morse complex has to be defined
including orientation spaces of unstable manifolds at the critical points. Alternatively in
the case of global quotients, we introduce the notion of weak group actions on Morse–
Smale–Witten complexes for non-invariant Morse–Smale functions on M, which give rise
to genuine group actions on the level of homology.

Keyword

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