I have this vague/ big picture question which might be too vague to answer (I don't mind it being deleted in this case). Do chain equivalences have anything to do with Leibniz product rule of differentiation:

Chain homotopy $D$ between two chain maps $a,b$: $$a_q-b_q=\delta_{q+2}D_{q+1}+D_q\delta_{q+1}$$

Leibniz product rule of differentiation ($f,g$ are some real value functions on a manifold $M$ and $X\in T_{m_0}M$:

$$X(fg)=f(m_0) X(g)+g(m_0)X(f)$$

I feel there is a similarity between the right hand sides of both equations, and I want to know if there is an explanation that could give me better intuition for chain equivalences.

  • $\begingroup$ You may read the answer here, where it is shown that a homotopy between two maps $f,g : M\to N$ defines a chain homotopy $h$. In the proof the Lie derivative and the Cartan magic formula is used. In another question, it is shown that the Cartan magic formula can be proved using a homotopy operator. $\endgroup$
    – user99914
    Mar 8, 2018 at 0:11
  • $\begingroup$ The cartan magic formula is not exactly the Leibniz rule though. $\endgroup$
    – user99914
    Mar 8, 2018 at 0:13


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