# Massey products and $A_{\infty}$ structures

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.

• Does the statement in Keller's reviews not work? webusers.imj-prg.fr/~bernhard.keller/publ/art.ps or webusers.imj-prg.fr/~bernhard.keller/publ/ainffun.pdf Feb 25, 2015 at 1:44
• @Aaron: Keller appears to be working over a field, where $H^i(C)$ is always free. Feb 25, 2015 at 1:57
• Not even the projection is well defined if the homology is not projective. Anyway, there is a nice notion of derived A-infinity algebra, due to Sagave, which works in general, although it is more complicated. I'd also say that what you find odd is typical phenomena. Feb 25, 2015 at 3:33

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.

• I really wish Tyler Lawson had notes covering DG-algebras and A$_\infty$-algebras for algebraic geometers wanting to learn homotopy theory... Feb 25, 2015 at 3:30
• @bananastack Keller has nice notes and introductory papers. Feb 25, 2015 at 3:35
• Can this be understood topologically? It looks like chains of the cartesian square of the Moore space $M(\mathbb Z/2,2)$... Feb 25, 2015 at 5:32
• @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 $\Bbb{RP}^\infty \times \Bbb{RP}^\infty$. Feb 25, 2015 at 17:46
• @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. Feb 25, 2015 at 17:49