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In general homology group of a complex is $Ker d/ Im d$(regardless of grading). However, in many case the square $d\circ d$ is not zero. For example, the study of Floer thoery needs A infinity structure, this structure is due to the geometric property.

Is there any general algebraic theory of the obsturction for the square of differential to be zero?

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    $\begingroup$ For most sources, a complex would have $d\circ d = 0$ so your terminology does seem a bit odd. What exactly is the definition of complex that you are using? Perhaps you need to work with deformation theoretic ideas. The A-infinity structure you mentioned is in a related area. $\endgroup$
    – Tim Porter
    Sep 6, 2014 at 11:10
  • $\begingroup$ The obstruction is the submodule $\operatorname{Im} (d\circ d)$. Without more context it seems difficult to say more. $\endgroup$
    – Mark Grant
    Sep 6, 2014 at 12:20
  • $\begingroup$ Funnily enough, even in the context of $A_\infty$-structures one has $d\circ d = 0$ (although $d$ is no differential, but a co-differential). Are you interested in some examples of obstructions, as well? $\endgroup$
    – Avitus
    Sep 8, 2014 at 9:54

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