Noncommutative rational homotopy type

Question

Ok, this question is much less ambitious than it might sound, but still:

Two commutative differential graded algebras (cdga's) are quasi-isomorphic if they can be connected by a chain of cdga quasi-isomorphisms. There is a similar definition for not necessarily commutative differential graded algebras (dga's).

1. If two $\mathbf{Q}$-cdga's, $A$ and $B$, are quasi-isomorphic as dga's, are they necessarily quasi-isomorphic as cdga's? I suspect that the answer is no, but don't know any counter-examples, nor can prove that such counter-examples exist.

2. Same question as 1 when $A$ and $B$ are the Sullivan $\mathbf{Q}$-polynomial cochain algebras of simply connected compact polyhedra. In other words, is the "rational noncommutative homotopy type" of compact simply-connected polyhedra the same as the usual rational homotopy type?

Answer 1

An affirmative answer (to both questions) appeared today on the arxiv, due to Campos, Petersen, Robert-Nicoud, Wierstra: https://arxiv.org/abs/1904.03585

Answer 2

Regarding Question 1: the case that one of the two algebras has vanishing differential is proven as one of the main theorems of the following paper of Bashar Saleh: "Noncommutative formality implies commutative and Lie formality", Algebr. Geom. Topol. Volume 17, Number 4 (2017), 2523-2542.

In other words, a cdga over $\mathbf Q$ which is formal as a dga is also formal as a cdga.

Answer 3

I can say that if A and B are commutative DGAs over Q which are connective (their homology groups are zero in negative degrees, using homological grading), then A and B are equivalent. The only proof I can come up with is a Postnikov argument based on the Hochschild-Kostant-Rosenberg theorem.

Namely, let P_nA and P_nB be Postnikov stages, which have the same homology as A and B respectively in degrees less than or equal to n and zero above; you can construct these by adding new polynomial generators to A and B with differentials that erase higher homology classes. Assume you've already constructed an equivalence from P_nA to P_nB.

Then P_n+1A and P_n+1B are classified by their next k-invariants. In the associative case this is an element in the Hochschild cohomology of P_nA with coefficients in a shift of the module M = H_n+1A = H_n+1B, and similarly for B. In the commutative case this is an element in the Andre-Quillen cohomology instead. These are classified by maps from the Hochschild homology (strictly speaking, its augmentation ideal) and Andre-Quillen homology (a shift of the derived Kahler differentials) to this shift of M.

Since we're in characteristic zero, the HKR theorem tells us that the Hochschild homology is the (derived) free graded-commutative algebra over P_nA on the Andre-Quillen homology. In particular, there is a retraction down. This implies that the collection of possible commutative k-invariants constructing an extension of P_nA by M are a subset of the possible associative k-invariants.

As a result, if P_n+1A and P_n+1B have a fixed equivalence as associative algebras, that implies that their associative k-invariants are equal. Hence that their commutative k-invariants are equal, so this allows us to lift to an equivalence of commutative algebras.

This is a sneaky problem. The fact that I needed to assume connectivity for this argument is pretty annoying, and I am not sure whether it is material or simply a failing on my part to get something general. Perhaps someone else can do better.

ADDED LATER: I believe that the same basic argument works in the coconnective case under added hypotheses (e.g. the algebra being simply connected, having only Q in degree 0 and 0 in degree 1). Unfortunately you cannot construct Postnikov stages quite as easily for general cochain algebras.

For example, let A be the commutative DGA Q[x,y,s,t] ⊗ Λ[w], a free algebra on generators x,y,s,t in degree zero and w in (cohomological) degree 1, satisfying dx = dy = dw = 0, ds = xw, dt = wy. Then there is a Massey product <x,w,y> = sy - xt in cohomology degree 0 which is nonzero, and any map from A to a DGA which has zero cohomology in positive degrees must necessarily send w to zero, and hence also this Massey product.

I have this vague feeling that in this simply-connected case (and also in the connective case) there might be a much shorter argument involving minimal models.

Answer 4

I think that the answer for the first question is "yes". I disagree with the Joey's argument because he uses a wrong definition of $Comm_\infty$ morphism. Let me recall the right definition first. A $Comm_\infty$ algebra $C$ is a coderivation $Q$ such that $Q^2=0$ on cofree LIE coalgebra cogenerated by $C[1]$. So, it should be a free Lie coalgebra, not COMMUTATIVE one. Now a $Comm_\infty$ morphism is just a map of from the two cofree Lie coalgebras which commutes with $Q$. It is a quis if it is, moreover, a quis of complexes.

Well, now suppose that $C$ and $D$ are two cdga placed in any degrees. If they are $A_\infty$ quis, it means that there is a map of cofree dg coalgebra cogenerated by $C[1]$ to the same guy cogenerated by $D[1]$. This is a map of coalgebras, then it defines a map between their primitive elements.

(Recall $x$ is primitive iff $\Delta x=1\otimes x+x\otimes 1$). Apriori this gives a map of free Lie coalgebras, but NOT dg Lie coalgebras. Moreover if $C$ and $D$ would be non-commutative, the primitive elements would be not closed under the differential. But if $C$ and $D$ are commutative, the maps of the Lie coalgebras of primitive elements is a map of dg Lie coalgebras.

It is only remains to note that this map of dg Lie coalgebras is quis; then it is by definition a $Comm_\infty$ quis map. But this easily follows from the PBW theorem. The tensor coalgebra as a vector space is the symmetyric (co)algebra of the free Lie coalgebra. If the map of Lie coalgebras would be not a quis, the corresponding map of their symmetric powers also would be not a quis. This would contradict that the initial map was an $A_\infty$ quis.

We are done.

Answer 5

As far as I know for positively graded connected and simply connected complexes with differential of degree $+1$ even in characteristic $0$ (corresponding to the rational homotopy theory case), this is an open problem.

So it would be nice to have a published paper that settles it.

Answer 6

I think the answer to question 1 is "Yes." Here's why:

Fix a field k of characteristic 0 (you asked about k = Q).

Let C and D be (graded-)commutative DG k-algebras. Then there exists a noncommutative quasi-isomorphism from C to D iff there exists an $A_{\infty}$ Quasi-iso $F:C \to D$. (In fact more is true---there is a one-to-one correspondence between equivalence classes of noncommutative quasi-iso's and equivalence classes of $A_\infty$ morphisms.)

Since there exists a commutative quasi-isomorphism from C to D iff there exists a $C_{\infty}$ quasi-iso $G: C \to D$, your question can be rephrased as follows: "Suppose C,D are commutative DG k-algebras, and $F: C \to D$ is an $A_{\infty}$ map. Does this imply the existence of a $C_{\infty}$ map?"

We can describe $A_{\infty}, C_{\infty}$ maps as follows: for a dga A, we can consider the cofree (conilpotent) coalgebra $T^c(A)$, and we observe that the dga structure maps $d: A \to A, \cdot: A \otimes A \to A$ define a map $T^c(A) \to A$ by projecting onto $A \oplus (A \otimes A)$ and then doing $d$ on the first factor and $\cdot$ on the second. This map can be extended as a coderivation $D_A:T^c(A) \to T^c(A)$, and the claim is that $D_A^2 = 0.$ An $A_{\infty}$ map $F:A \to B$ is by a definition a dg-coalgebra map $F:(T^c(A), D_A) \to (T^c(B), D_B)$. Similarly, a commutative dga gives rise to a cofree cocommutative dg coalgebra, using $S^c$, and $C_{\infty}$ morphisms are dg-coalgebra maps between these. An $C_{\infty}$ map is actually an $A_{\infty}$ map with some symmetry properties (since $S^c \hookrightarrow T^c$).

Here's a first step in the argument "up to signs" that you can symmetrize the $A_{\infty}$ map F.

To get a $C_{\infty}$ map from F, write $F = f + f_2 + f_3 + \cdots$, where $f_n : C^{\otimes n} \to D$. Then the condition that $F$ is an $A_{\infty}$ morphism gives $(df_2)(x,y) = f(x) \cdot_{D} f(y) \pm f(x \cdot_{C} y)$ where $df_2$ should be understood as the differential in the Hom complex $Hom(C^{\otimes 2}, D)$. Since the products in C and D are commutative, we actually get: $(df_2)(x,y) = f(x) \cdot_{D} f(y) \pm f(x \cdot_{C} y) = \pm f(y) \cdot_D f(x) \pm f(y \cdot_C x) = (df_2)(y,x)$ so we see that at least $(df_2)$ is graded symmetric. Since k has characteristic 0, we can define

$g_2 := \frac{1}{2} \left( f_2 + f_2^{op} \right)$

and since

$(dg_2) = \frac{1}{2} d \left( f_2 + f_2^{op} \right) = \frac{1}{2} \left( df_2 + d(f_2^{op}) = \frac{1}{2}( df_2 \pm (df_2)^{op} ) = (df_2) \right)$

we see that $dg_2$ still provides the homotopy between $f( \cdot_C )$ and $f( ) \cdot f( )$.

I think an argument along these lines will also provide suitable $g_n$ for $n \geq 3$.

I wonder if someone can provide an "abstract nonsense" answer along the following lines: the functor i: Commutative DGAs -----> nonCommutative DGAs has a left adjoint which preserves quasi-iso's (does abelianization preserve maps inducing isomorphisms in Homology?), and so we can verify that the image of Commutative DGAs in (nonCommutative DGAs)$[W^{-1}]$ has the universal property that a localization (Commutative DGAs)$[W^{-1}]$ should have... (Where $W$ above should be taken to be the collection of maps inducing iso's in homology in each category).

Answer 7

I think I was wrong in my answer. The matter is that I have a map of dg associative coalgebras, it indeed induces a map on primitive elements, but these primitive elements are the cogenerators, that is elements of $A[1]$, not of $Lie(A[1])$. So I can not deduce that I have an induced map of free dg Lie coalgebras. I do not know is it the case or not, but my speculation certaily fails.

Dear James, concerning your question 2). Here is not any mistake. The difference with bar complex case is that it is not tensor power of generators as COMPLEX. In commutative case, the free Lie (co)algebra is closed under the differential, and the entire complex is the symmetric power (by PBW) of this Lie (co)algebra AS A COMPLEX. That is the only component of the differential comes from this free dg Lie (co)algebra, and we have the symmetric power of a complex. So the claim reads like: if two complexes are quis, their symmetric powers also are.

Answer 8

This is an attempt to rephrase some of Joey's answer and also to add in the higher operations required.

Suppose we have two cdgas A and C and that they're connected by an $A_\infty$ morphism. This is encoded as a map from $BA$, the associative bar construction on A to C. A $C_\infty$ morphism is encoded by a morphism from the commutative bar construction, $B_cA$, this takes the form of a quasi-free colie algebra on A. There is a natural transformation from the free coassociative algebra functor to the free colie algebra functor and this coincides with a natural transformation from the associative to commutative bar functor. This is ofcourse the wrong direction, the natural transformation says "every $C_\infty$ morphism is an $A_\infty$ morphism".

To go the other way we observe that the maps $BA\rightarrow B_cA$ split. I wont offer a proof here but the proof goes by constructing certain idempotents, it's known as the Hodge decomposition of the bar complex of a commutative algebra. We now have the composition

$B_cA\rightarrow BA \rightarrow C$

which is our $C_\infty$ morphism. Warning The $A_\infty$ morphism we get from this $C_\infty$ morphism is not necessarily the one we started off with, we have just stripped away all the "stuff" coming from other parts of a Hodge filtration.

Finally I'd like to add that your intuition was broadly right, there will be examples of non char 0 cases where the answer to your question will be "no".