Morphisms of Hochschild (or cyclic) homology induced by homotopic maps Let $f$ and $g$ be two maps between DG algebras $A$ and $B$, and assume that $f$ and $g$ are homotopic as chain maps, hence they induce the same map on the level of homology. Moreover, $f$ and $g$ induce morphisms between $\mathrm{HH}_\bullet(A)$ and $\mathrm{HH}_\bullet(B)$ and $\mathrm{HC}_\bullet(A)$ and $\mathrm{HC}_\bullet(B)$. What can we say about those morphisms between Hochschild or cyclic homology?
Everything is over a field of characteristic zero.
Any suggestions and helps would be really appreciated.
 A: $\newcommand{\dd}{\mathrm d}\DeclareMathOperator{\Sym}{Sym}\DeclareMathOperator{\id}{id}$If $f,g:A\to B$ are dga morphisms which are homotopic as chain maps, the maps induced by $f$ and $g$ on Hochschild homology and its variants can be different. I give an example above the line; the easiest way to construct it, at least for me, is to systematically investigate the (homotopy) functoriality of Hochschild homology, which I sketch below the line.
Let $A = \mathbb k[\{x_k\}_{k\ge 1}]$ be the free nonunital graded algebra on generators $x_k$ in (cohomological) degree $1$, equipped with the differential $\dd x_k = \sum_{i+j=k} x_ix_j$. In other words, $A = \Omega(B(\mathbb k[x]/x^2))$ is the cobar-bar resolution of the free nonunital graded commutative algebra on one generator in degree $1$; in particular there are chain maps
\begin{align*}
r: A &\leftrightarrows H_*(A)\cong \mathbb k[x]/x^2:i\\
x_k &\mapsto \begin{cases}x&\text{ for }k=1\\ 0&\text{ else}\end{cases}\\
x_1 &\leftarrow\!\shortmid x
\end{align*}
such that $r$ is a dga morphism, $r\circ i = \id_{H_*(A)}$ and there is a chain homotopy $h$ between $\id_A$ and $i\circ r$.
Let $f:A\to H_*(A)$ be the zero map, and let $g:A\to H_*(A)$ be the (nonunital) algebra map with $g(x_k) = \begin{cases}x&\text{ for }k = 2\\0&\text{ else}\end{cases}$. Both are chain maps since the target is concentrated in degree $1$, and since $g\circ i = 0$, the map $g\circ h$ is a chain homotopy between $f$ and $g$.
Let $\alpha = x_1\otimes x_1\in C_1(A)$. Since $x$ is in odd degree, the Koszul sign rule gives $\dd\alpha = 2x_1^2$, so that $\beta = x_1\otimes x_1 - 2x_2\in C_1(A)$ is closed. Its image under $f$ vanishes, while its image under $g$ gives the cycle $2x\in HH_1(H_*(A))$, which is nonzero since it maps to the nonzero element $2x$ under the trace map $HH_1(H_*(A))\to H_*(A)$ arising from the commutativity of $H_*(A)$. Thus $f$ and $g$ do not induce the same map on Hochschild homology.

Hochschild homology is a functor for $A_\infty$-morphisms, i.e. a dga morphism $\Omega(B(A))\to B$ (explicitly, a collection of maps $f_n:A^{\otimes n}\to B[1-n]$ satisfying certain quadratic equations) induces a chain map $HH_*(A)\to HH_*(B)$. Said differently, they are Maurer-Cartan elements of the dgla $\operatorname{Hom}(B(A),B)$ (the Maurer-Cartan equation is exactly the quadratic equation satisfied by the components $f_n$). Now for any (nilpotent) dgla $\mathfrak g$, the set of Maurer-Cartan elements form the $0$-simplicies of a Kan complex/homotopy type with $n$-simplices given by Maurer-Cartan elements of $\mathfrak g\otimes\Omega_{PL}^*(\Delta^n)$, and we can actually replace the cdga $\Omega_{PL}^*(\Delta^n)$ by a smaller $C_\infty$-algebra, compare Robert-Nicoud--Vallette's Higher Lie Theory. In particular, we now have a straightforward definition of a homotopy between two $A_\infty$-morphisms: It is essentially a family of $A_\infty$-morphisms $f_t$, parametrized by $t\in [0,1]$, together with a "coherent nullhomotopy" of $\partial_t f_t$, such that $f_0 = f,f_1 = g$.
Very explicitly, given the two morphisms $f,g$, it is a collection of maps $h_n:A^{\otimes n}\to B[-n]$ which satisfies a quadratic identity involving the components of $f$ and $g$, which for small $n$ and strict morphisms $f,g$ (i.e. $f_n = g_n = 0$ for $n>1$) becomes

*

*$h_1$ is a chain homotopy between $f_1$ and $g_1$

*$h_2$ is a chain homotopy between $\mu_B\circ(f_1\otimes h_1 + h_1\otimes g_1)$ and $h\circ \mu_A$

*$h_3$ is a chain nullhomotopy of a map $A^3\to B$ of degree $-2$, defined in terms of the maps $f_1,g_1,h_1,h_2$
Explicit formulas and a generalization to higher homotopies can be found in Mazuir's Higher algebra of A∞ and ΩBAs-algebras in Morse theory II.
Your assumption that $f$ and $g$ are chain homotopic is the first step in this series of equations; each subsequent step asks you to find a nullhomotopy for a chain map defined in terms of the previous choices. In particular, this is impossible if this chain map is not nullhomotopic. This makes it easy to find an example of such a map; it is slightly more work to find one such that the induced maps on Hochschild homology are different:
Let $V$ be a graded vector space and $A_V = \Omega((\Sym^* V[1])[-1])$ be the Koszul resolution of the free graded commutative algebra $\Sym^*V$, i.e. $A_V$ is the free algebra on $\Sym^* V[1]$ with the differential on generators defined via the coalgebra structure. By the HKR theorem, we have $HH_*(A_V)\cong HH_*(\Sym^*V)\cong \Sym^* (V\oplus V[1])$. The Hochschild cohomology of $A_V$ is quasiisomorphic to $\Sym^* (V\oplus V^\vee[-1])$; essentially, it gives infinitesimal deformations of the identity as a $A_\infty$-automorphism of $\Sym^* V$. In particular, any polyvector field $X\in\Sym^* V\otimes\Sym^{\ge 2}V^\vee[-1]$ of cohomological degree $0$ integrates to a $A_\infty$-automorphism of $\Sym^* V$ whose linear component is the identity, which can then be lifted to a dga automorphism of $A_V$ whose underlying chain map is homotopic to the identity. Its action on Hochschild homology is (the exponentiation of) the "Lie derivative" action of polyvector fields on differential forms, and thus usually not equal to the identity map.
The example above essentially arises in this way, except for the fact that I took a deformation of the zero map. The polyvector field is $\partial_x\wedge\partial_x$, which defines an $A_\infty$-morphism from $H_*(A)$ to itself with nonvanishing degree $2$ component, and the differential form is $x\dd x$. My impression is that the explicit computation does not do much to explain why the example works. If you are interested in the functoriality properties of Hochschild and cyclic homology from a more abstract point of view, I can recommend Krause--Nikolaus's lecture series on topological cyclic homology.
