I know the general theorem of Kadeishvili which says that, for a DGA $C$, when $H^{i}(C)$, $i\geq 0$, is free, $H(C)$ can be made into an $A_{\infty}$ algebra. If my understanding is correct, the proof essentially uses the freeness of $H^{i}(C)$ to produce a section $$s:H(C)\to C$$ of the projection $p:C\to H(C)$. Then a choice of chain homotopy between the identity and $sp$ can be used to define the maps $$m_{i}:\otimes^{i}H(C)\to H(C).$$ It is not hard to see that these give you an element of the higher Massey products (when defined).

Here is my question: When $H^{i}(C)$ is not free, is there any good way to define an $A_{\infty}$ structure on $H(C)$? It seems that the fact that one has an actual chain homotopy is crucial to the construction and in general $p$ may only be a quasi-iso. I hope this question is not too elementary (I am still a lowly graduate student). It just seems odd to me that the higher Massey products can be defined in general, even for torsion $H(C)$ (as long as the lower ones vanish) but one may not be able to identify these with a full $A_{\infty}$ structure.


Let $C = \Bbb Z[x,y] \otimes \Lambda[u,v]$, with $x$ and $y$ in (homological) degree 2 and $u$ and $v$ in degree 3, with $dx = dy = 0$ and $du = 2x$, $dv = 2y$. Then $H_5(C)$ is $\Bbb Z/2$, generated by the Massey product $x v - u y = \langle x,2,y\rangle$ (with no indeterminacy in this case).

This structure does not come from the homology equipped with an $A_\infty$ structure, because then we would have $$\langle x,2,y \rangle = m_3(x \otimes 2 \otimes y) = 2 \cdot m_3(x \otimes 1 \otimes y) = 0.$$

The usual generalization clarifies that the original could be viewed as, instead of a theorem about homology, a theorem about chain equivalence: if $C \to D$ is a chain homotopy equivalence and $C$ is a DGA, then $D$ gets an $A_\infty$ structure.

| cite | improve this answer | |
  • 1
    $\begingroup$ I really wish Tyler Lawson had notes covering DG-algebras and A$_\infty$-algebras for algebraic geometers wanting to learn homotopy theory... $\endgroup$ – bananastack Feb 25 '15 at 3:30
  • 2
    $\begingroup$ @bananastack Keller has nice notes and introductory papers. $\endgroup$ – Fernando Muro Feb 25 '15 at 3:35
  • 2
    $\begingroup$ Can this be understood topologically? It looks like chains of the cartesian square of the Moore space $M(\mathbb Z/2,2)$... $\endgroup$ – მამუკა ჯიბლაძე Feb 25 '15 at 5:32
  • $\begingroup$ @მამუკაჯიბლაძე I'm not sure if there's an H-space with this chain algebra (the fact that all elements graded-commute seems to make that a little sensitive). If you switch $x$ and $y$ to cohomological degree $2$ and $u$ and $v$ to cohomological degree $1$, then I believe that this is equivalent to cochains on $\Bbb{RP}^\infty \times \Bbb{RP}^\infty$. $\endgroup$ – Tyler Lawson Feb 25 '15 at 17:46
  • $\begingroup$ @bananastack That's very flattering. I think that one of my main difficulties is not knowing exactly where most algebraic geometers have their strongest points of interest about these structures, and there are others who seem to understand that much better. $\endgroup$ – Tyler Lawson Feb 25 '15 at 17:49

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.