## Formality of a higher-codimensional Swiss-cheese operad.(English)Zbl 1495.55005

The little disks operads $$\mathrm{D}_n$$ are classical objects in algebraic topology that were first introduced to study iterated loop spaces. The elements of $$\mathrm{D}_n(k)$$ consist of configurations of $$k$$ disks inside the unit $$n$$-disk $${\mathbb{D}}^n$$. An important property of $$\mathrm{D}_n$$ is its formality. The Swiss-cheese operads $$\mathrm{SC}_n$$, $$n\geq 2$$, are relative versions of the little disks operads. Unlike the little disks operads, the Swiss-cheese operads are not formal.
The author introduces higher-codimensional variants of the Swiss-cheese operads, $$\mathrm{CD}_{mn}$$, where $$n-2\geq m\geq 1$$, called the complementarily constrained disks operads. On the level of homology, these operads encode actions of $$\mathrm{D}_n$$-algebras on $$\mathrm{D}_m$$-algebras by central derivations. The elements of $$\mathrm{CD}_{mn}(k,l)$$ are given by configurations of $$k$$ disks centered on $${\mathbb{D}}^m\subset {\mathbb{D}}^n$$ and $$l$$ disks entirely contained in the complement $${\mathbb{D}}^n\setminus {\mathbb{D}}^m$$.
The main result of the paper states that $$\mathrm{CD}_{mn}$$ is formal over $${\mathbb{R}}$$. The proof is inspired by the proofs of the formality of the little disks operad by M. Kontsevich [Lett. Math. Phys. 48, No. 1, 35–72 (1999; Zbl 0945.18008)] and by P. Lambrechts and I. Volić [Formality of the little $$N$$-disks operad. Providence, RI: American Mathematical Society (AMS) (2014; Zbl 1308.55006)]. The motivation for the article comes from the study of the configuration spaces of the complement $$N\setminus M$$, where $$N$$ is a closed $$n$$-manifold and $$M$$ is a closed $$m$$-submanifold of $$N$$ of codimension $$\geq 2$$.

### MSC:

 55P48 Loop space machines and operads in algebraic topology 55R80 Discriminantal varieties and configuration spaces in algebraic topology

### Citations:

Zbl 0945.18008; Zbl 1308.55006
Full Text:

### References:

 [1] ; Arnold, V. I., The cohomology ring of the group of dyed braids, Mat. Zametki, 5, 227 (1969) [2] 10.5802/aif.2924 · Zbl 1329.57035 [3] ; Axelrod, Scott; Singer, I. M., Chern-Simons perturbation theory, II, J. Differential Geom., 39, 1, 173 (1994) · Zbl 0827.53057 [4] 10.1007/s00029-016-0242-1 · Zbl 1365.57037 [5] 10.1090/S0002-9904-1968-12070-1 · Zbl 0165.26204 [6] 10.1007/BFb0080467 [7] 10.1007/BF01589496 · Zbl 0842.14038 [8] 10.1016/j.jalgebra.2015.01.032 · Zbl 1327.17019 [9] 10.1016/j.jpaa.2014.06.010 · Zbl 1305.18032 [10] 10.2140/agt.2016.16.1683 · Zbl 1355.55007 [11] 10.5802/ambp.219 · Zbl 1141.55006 [12] 10.1090/surv/217.2 · Zbl 1375.55007 [13] 10.1515/gmj-2018-0061 · Zbl 1408.18017 [14] ; Fresse, Benoit, Little discs operads, graph complexes and Grothendieck-Teichmüller groups, Handbook of homotopy theory, 405 (2020) · Zbl 1476.55030 [15] 10.1007/s40062-018-0198-2 · Zbl 1405.18015 [16] 10.4171/JEMS/961 · Zbl 1445.18014 [17] 10.2307/2946631 · Zbl 0820.14037 [18] 10.4007/annals.2009.170.271 · Zbl 1246.17025 [19] 10.1007/BF02699127 · Zbl 0678.53059 [20] 10.1007/978-3-642-71714-7 [21] 10.2140/agt.2011.11.2477 · Zbl 1254.14066 [22] 10.2140/agt.2013.13.2039 · Zbl 1272.18006 [23] 10.1016/j.topol.2016.03.023 · Zbl 1368.55003 [24] 10.1007/s00222-018-0842-9 · Zbl 1422.55031 [25] 10.1007/978-3-0348-9112-7_5 · Zbl 0872.57001 [26] 10.1023/A:1007555725247 · Zbl 0945.18008 [27] 10.1023/B:MATH.0000027508.00421.bf · Zbl 1058.53065 [28] ; Lambrechts, Pascal; Volić, Ismar, Formality of the little N-disks operad. Mem. Amer. Math. Soc., 1079 (2014) · Zbl 1308.55006 [29] 10.1112/jtopol/jtv018 · Zbl 1333.55009 [30] 10.1007/978-3-642-30362-3 · Zbl 1260.18001 [31] 10.1006/jabr.1998.7709 · Zbl 0949.58009 [32] 10.1007/BFb0067491 [33] 10.1007/s00029-013-0135-5 · Zbl 1312.55008 [34] 10.2140/agt.2002.2.1001 · Zbl 1024.57015 [35] 10.1090/ulect/037 [36] 10.1007/978-3-0348-8312-2_23 · Zbl 1034.55007 [37] 10.1007/s00029-017-0382-y · Zbl 1393.53092 [38] 10.1007/s00029-004-0381-7 · Zbl 1061.55013 [39] ; Sinha, Dev P., The (non-equivariant) homology of the little disks operad, OPERADS 2009. Sémin. Congr., 26, 253 (2013) · Zbl 1277.18012 [40] 10.4310/HHA.2011.v13.n2.a16 · Zbl 1231.55009 [41] 10.1023/B:MATH.0000017651.12703.a1 · Zbl 1048.18007 [42] 10.1353/ajm.2018.0006 · Zbl 1428.18033 [43] 10.2140/agt.2020.20.1431 · Zbl 1441.55007 [44] 10.1090/conm/239/03610 · Zbl 0946.55005 [45] 10.1007/s00222-014-0528-x · Zbl 1394.17044 [46] 10.1215/00127094-3450644 · Zbl 1346.53077
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.