# Dave Hartman

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 Jul2 awarded Curious Jan29 awarded Nice Question Oct14 accepted Two-sided bar construction Oct12 comment Two-sided bar construction If $A$ is just an algebra, then $HH_*(A)$ "measures" how far $A$ is from being a flat $(A\otimes A^{op})$-module, right? Oct12 comment Two-sided bar construction Thank you, I wasn't aware of these problems. I was under the impression, that the case when $A$ is a differential, graded algebra would be similar to the case when $A$ is just an algebra. In this case the two-sided bar construction is a free resolution, but apparently the situation is much more complicated in case $A$ is graded, differential...which is a shame, since I now have no idea how to interpret Hochschild homology of a DG-Algebra. What does Hochschild homology of a DG-algebra "measures"? Oct12 revised Two-sided bar construction added 15 characters in body Oct12 comment Two-sided bar construction Thank you very much for this nice explanation. How does $(B=B(A,A,A),d)$ give rise to a free $A\otimes A^{op}$-resolution of $A$? Does one just consider $(B,(-1)^pd^v)$, where $B$ is now only graded by the homological degree? I would like to get to the Hochschild homology $Tor_*^{A\otimes A^{op}}(A,A)$ of $A$ via the resolution arising from the two-sided bar construction. Oct12 comment Two-sided bar construction But then I don't understand how $(B(A,A,A),d)$ is a chain complex giving a free resolution of $(A,d_A)$ as an $A\otimes A^{op}$-module. Or is this not true and what is really going on is that $(B(A,A,A),d_1)$ is a chain complex (graded by wordlength on $T(s\bar{A})$) giving a free resolution of $(A,d_A)$ as an $A\otimes A^{op}$-module? Oct12 revised Two-sided bar construction added 9 characters in body Oct12 asked Two-sided bar construction Jun24 accepted Is $C^\nu(X,Y)$ a Banach manifold and a Lindelöf space? Jun17 comment Is $C^\nu(X,Y)$ a Banach manifold and a Lindelöf space? @Peter Michor: Thank you for elaborating. Are smooth bump functions necessary in order to get a local addition? Does my space $Y$ admit a local addition? Jun12 comment Is $C^\nu(X,Y)$ a Banach manifold and a Lindelöf space? @Peter Michor: Thank you for the additional reference. Yes, $Y$ is an open set in a Banach space. In fact, $Y=\mathbb{R}^2\times F_2(Gvect(f))$ which is an open set in $\mathbb{R}^2\times G_1^2$. $G_1$ is the space from your answer here: mathoverflow.net/questions/127843/… . But I consider $G_1$ with the $C^k$ (instead of $C^\infty$) topology so that it is a Banach space (instead of a Fréchet space). Jun11 revised Is $C^\nu(X,Y)$ a Banach manifold and a Lindelöf space? added 18 characters in body Jun8 revised Is $C^\nu(X,Y)$ a Banach manifold and a Lindelöf space? added 1 characters in body Jun7 comment Is $C^\nu(X,Y)$ a Banach manifold and a Lindelöf space? @Peter Michor: Thank you for the edit. Although I don't understand why it is true, I'm very glad to hear that $C^\nu(X,Y)$ is separable if $X$ is compact and $Y$ is separable! I'd appreciate it very much if you could explain in some more detail why this is in fact true (or point me to a reference). Unfortunately I don't know what the "completed inductive tensor product" is (and I didn't find much about it on the internet...). Thank you for your help. Jun7 revised Is $C^\nu(X,Y)$ a Banach manifold and a Lindelöf space? added 14 characters in body Jun5 revised Is $C^\nu(X,Y)$ a Banach manifold and a Lindelöf space? added 1 characters in body Jun5 comment Is $C^\nu(X,Y)$ a Banach manifold and a Lindelöf space? Thank you very much for your answer and the references! From your answer I gather that the existence of such a local addition is not a very restrictive condition on $Y$, is this correct? Do you know if $C^\nu(X,Y)$ is a Lindelöf space (or second countable)? Jun5 asked Is $C^\nu(X,Y)$ a Banach manifold and a Lindelöf space?