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Preliminaries: Let $(A,\partial)$ be a differential graded $k$-algebra with an augmentation $\epsilon$. That is, $\epsilon$ is a DGA map $\epsilon:(A,\partial) \to k$ where $k$ is given the trivial DGA structure. With this setup one can form the linearized chain complex $L(A,\epsilon) := \text{ker}(\epsilon)/\text{ker}(\epsilon)^2$, which has a well-defined differential $\partial_\epsilon$ induced by $\partial$. The homology $LH(A,\partial;\epsilon)$ of $L(A,\epsilon)$ is called the linearized homology of the DGA $(A,\partial)$ with respect to $\epsilon$.

It is simple to check that if $f:A \to B$ is a DGA map and $\epsilon:B \to k$ is an augmentation, then there is an induced map: $$f_*:LH(A,\partial;\epsilon \circ f) \to LH(B,\partial;\epsilon)$$ Now, one would like to say that there is some sense in which homotopic DGA maps induce the same map on linearized homology. Here is an example of the kind of homotopy invariance statement that I have in mind:

Fake (?) Lemma: If $f,g:(A,\partial) \to (B,\partial)$ are homotopic, i.e. there is a linear chain homotopy $h$ such that: $$h \partial + g \partial = f - g$$ then there is a natural isomorphism $LH(A,\partial;\epsilon \circ f) \simeq LH(A,\partial;\epsilon \circ g)$ such that the following diagram commutes: $$ \begin{CD} LH(A,\partial;\epsilon \circ f) @>\simeq>> LH(A,\partial;\epsilon \circ g)\\ @V f_* V V @VV g_* V\\ LH(B,\partial;\epsilon) @>>=> LH(B,\partial;\epsilon) \end{CD} $$

However, it isn't obvious to me how to prove this or if it's true as stated, and in fact the references that I have looked at suggest that it's false (see the motivation below for some reference pointers). My question is thus the following.

Question: What is the right notion of chain homotopy between DGAs such that homotopic DGA maps induce the same map on linearized homology (perhaps in the sense of my fake lemma)?

Motivation: My motivation comes from understanding contact homology.

Pardon recently defined full contact homology in [1] and shown that it's well-defined. This involves showing that homotopic exact symplectic bordisms between two contact manifolds induce chain homotopic maps of the contact DGAs associated to the contact manifolds at either end. However, on p. 14 he remarks that the version of chain homotopy that he has established is not strong enough to prove well-definedness of linearized contact homology. This is an invariant that is constructed from the DGA defining contact homology using a linearization procedure, like above.

Pardon suggests some papers that describe how to construct homotopies with the right properties (see [2] or [3]). However, I failed to dig out the answer to my question from those references.

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  • $\begingroup$ Between semi-free connected DGAs, quasi-isomorphism will suffice, as linearisation is a left Quillen functor. $\endgroup$ – Jon Pridham Apr 30 '18 at 5:34
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    $\begingroup$ Perhaps I can point out that what you call the "linearized chain complex" is often called the "indecomposables" of the DGA. It preserves quasi-isomorphisms of cofibrant DGAs as @JonPridham points out. If you take the left derived functor of "linearization", you get the classical notion of Hochschild homology. In my mind (but I'm no expert on contact homology) it's not very reasonable to expect indecomposables to preserve homotopy, that's why we have to take the derived functor. $\endgroup$ – Najib Idrissi Apr 30 '18 at 8:36

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