## Return to Answer

2 edited body

Hi,

Your Question:When are the dg-Lie algebra structures on Hochschild cochains: HCH∗(C∗(Ω(M),Q),C∗(Ω(M),Q))[1]≅HCH∗(C∗(M,Q),C∗(M,Q))[1] quasi-isomorphic ?

this is always true.

Step 1: From my paper with Felix and Thomas, looking at the proof, you can see that dg-Lie algebra structures on Hochschild cochains: $HCH∗(\Omega C_*(M),\Omega C_*(M))[1]≅HCH∗(C∗(M,Q),C∗(M,Q))[1]$ are quasi-isomorphic Here $\Omega C_*(M)$ is the Adams Cobar construction on the coalgebra C_*(M).

Step 2: There is an quasi-isomorphism of chains algebras called Adams cobar equivalence $\Theta:\Omega C_*(M)\rightarrow C∗(Ω(M)$. In our paper, we prove (very short proof) that this quasi-isomorphism $\Theta$ induces an isomorphism of Gerstenhaber algebras between $HH∗(C∗(Ω(M),Q),C∗(Ω(M),Q))$ and $HH∗(\Omega C_*(M),\Omega C_*(M))$. In particular, we have an isomorphism of graded Lie algebras. You want a dg-Lie algebra isomorphism on Hochschild cochains: HCH∗(C∗(Ω(M),Q),C∗(Ω(M),Q))[1] and HCH∗(\Omega C_(M),\Omega C_(M)). This is true. One of my coauthor had a proof. But it is not in our paper, since I thought he it was not interesting and too complicated. But if I remember well, Hamilton and Lazarev proved it in a paper following our paper. I think that Keller proved also in the paper you quote "Derived Invariance of Higher Structures of the Hochschild complex".

ps: There is two versions of my paper with Felix and Thomas, the published squezeed version valid only over a field, and the arxiv longer version with more details.

1

Hi,

Your Question:When are the dg-Lie algebra structures on Hochschild cochains: HCH∗(C∗(Ω(M),Q),C∗(Ω(M),Q))[1]≅HCH∗(C∗(M,Q),C∗(M,Q))[1] quasi-isomorphic ?

this is always true.

Step 1: From my paper with Felix and Thomas, looking at the proof, you can see that dg-Lie algebra structures on Hochschild cochains: $HCH∗(\Omega C_*(M),\Omega C_*(M))[1]≅HCH∗(C∗(M,Q),C∗(M,Q))[1]$ are quasi-isomorphic Here $\Omega C_*(M)$ is the Adams Cobar construction on the coalgebra C_*(M).

Step 2: There is an quasi-isomorphism of chains algebras called Adams cobar equivalence $\Theta:\Omega C_*(M)\rightarrow C∗(Ω(M)$. In our paper, we prove (very short proof) that this quasi-isomorphism $\Theta$ induces an isomorphism of Gerstenhaber algebras between $HH∗(C∗(Ω(M),Q),C∗(Ω(M),Q))$ and $HH∗(\Omega C_*(M),\Omega C_*(M))$. In particular, we have an isomorphism of graded Lie algebras. You want a dg-Lie algebra isomorphism on Hochschild cochains: HCH∗(C∗(Ω(M),Q),C∗(Ω(M),Q))[1] and HCH∗(\Omega C_(M),\Omega C_(M)). This is true. One of my coauthor had a proof. But it is not in our paper, since I thought he was not interesting and too complicated. But if I remember well, Hamilton and Lazarev proved it in a paper following our paper. I think that Keller proved also in the paper you quote "Derived Invariance of Higher Structures of the Hochschild complex".

ps: There is two versions of my paper with Felix and Thomas, the published squezeed version valid only over a field, and the arxiv longer version with more details.
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 12, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0}, "config": {"markdown_headings": true, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "github"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "english_confidence": 0.8872004747390747, "perplexity_flag": "middle"}