| changeset 450 | 56a31852242e |
| parent 449 | ae5fd0a7a8a3 |
| child 534 | 2b1d52c41ac5 |
| 449:ae5fd0a7a8a3 | 450:56a31852242e |
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6 about the action of the little disks operad on Hochschild cochains. |
6 about the action of the little disks operad on Hochschild cochains. |
7 The first several paragraphs lead up to a precise statement of the result |
7 The first several paragraphs lead up to a precise statement of the result |
8 (Theorem \ref{thm:deligne} below). |
8 (Theorem \ref{thm:deligne} below). |
9 Then we give the proof. |
9 Then we give the proof. |
10 |
10 |
11 \nn{Does this generalization encompass Kontsevich's proposed generalization from \cite[\S2.5]{MR1718044}, |
11 %\nn{Does this generalization encompass Kontsevich's proposed generalization from \cite[\S2.5]{MR1718044}, |
12 that (I think...) the Hochschild homology of an $E_n$ algebra is an $E_{n+1}$ algebra? -S} |
12 %that (I think...) the Hochschild homology of an $E_n$ algebra is an $E_{n+1}$ algebra? -S} |
13 |
13 |
14 %from http://www.ams.org/mathscinet-getitem?mr=1805894 |
14 %from http://www.ams.org/mathscinet-getitem?mr=1805894 |
15 %Different versions of the geometric counterpart of Deligne's conjecture have been proven by Tamarkin [``Formality of chain operad of small squares'', preprint, http://arXiv.org/abs/math.QA/9809164], the reviewer [in Confˇrence Moshˇ Flato 1999, Vol. II (Dijon), 307--331, Kluwer Acad. Publ., Dordrecht, 2000; MR1805923 (2002d:55009)], and J. E. McClure and J. H. Smith [``A solution of Deligne's conjecture'', preprint, http://arXiv.org/abs/math.QA/9910126] (see also a later simplified version [J. E. McClure and J. H. Smith, ``Multivariable cochain operations and little $n$-cubes'', preprint, http://arXiv.org/abs/math.QA/0106024]). The paper under review gives another proof of Deligne's conjecture, which, as the authors indicate, may be generalized to a proof of a higher-dimensional generalization of Deligne's conjecture, suggested in [M. Kontsevich, Lett. Math. Phys. 48 (1999), no. 1, 35--72; MR1718044 (2000j:53119)]. |
15 %Different versions of the geometric counterpart of Deligne's conjecture have been proven by Tamarkin [``Formality of chain operad of small squares'', preprint, http://arXiv.org/abs/math.QA/9809164], the reviewer [in Confˇrence Moshˇ Flato 1999, Vol. II (Dijon), 307--331, Kluwer Acad. Publ., Dordrecht, 2000; MR1805923 (2002d:55009)], and J. E. McClure and J. H. Smith [``A solution of Deligne's conjecture'', preprint, http://arXiv.org/abs/math.QA/9910126] (see also a later simplified version [J. E. McClure and J. H. Smith, ``Multivariable cochain operations and little $n$-cubes'', preprint, http://arXiv.org/abs/math.QA/0106024]). The paper under review gives another proof of Deligne's conjecture, which, as the authors indicate, may be generalized to a proof of a higher-dimensional generalization of Deligne's conjecture, suggested in [M. Kontsevich, Lett. Math. Phys. 48 (1999), no. 1, 35--72; MR1718044 (2000j:53119)]. |
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190 \stackrel{\id\ot\alpha_1}{\to} \bc_*(R_1\cup N_1) |
190 \stackrel{\id\ot\alpha_1}{\to} \bc_*(R_1\cup N_1) |
191 \stackrel{f_1}{\to} \bc_*(R_2\cup M_2) \stackrel{\id\ot\alpha_2}{\to} |
191 \stackrel{f_1}{\to} \bc_*(R_2\cup M_2) \stackrel{\id\ot\alpha_2}{\to} |
192 \cdots \stackrel{\id\ot\alpha_k}{\to} \bc_*(R_k\cup N_k) |
192 \cdots \stackrel{\id\ot\alpha_k}{\to} \bc_*(R_k\cup N_k) |
193 \stackrel{f_k}{\to} \bc_*(N_0) |
193 \stackrel{f_k}{\to} \bc_*(N_0) |
194 \] |
194 \] |
195 (Recall that the maps $\id\ot\alpha_i$ were defined in \S\ref{ss:module-morphisms}s.) |
195 (Recall that the maps $\id\ot\alpha_i$ were defined in \S\ref{ss:module-morphisms}.) |
196 \nn{need to double check case where $\alpha_i$'s are not closed.} |
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197 It is easy to check that the above definition is compatible with the equivalence relations |
196 It is easy to check that the above definition is compatible with the equivalence relations |
198 and also the operad structure. |
197 and also the operad structure. |
199 We can reinterpret the above as a chain map |
198 We can reinterpret the above as a chain map |
200 \[ |
199 \[ |
201 p: C_0(FG^n_{\ol{M}\ol{N}})\ot \hom(\bc_*(M_1), \bc_*(N_1))\ot\cdots\ot\hom(\bc_*(M_k), \bc_*(N_k)) |
200 p: C_0(FG^n_{\ol{M}\ol{N}})\ot \hom(\bc_*(M_1), \bc_*(N_1))\ot\cdots\ot\hom(\bc_*(M_k), \bc_*(N_k)) |