{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,28]],"date-time":"2026-08-28T08:20:09Z","timestamp":1787905209199,"version":"build-2784847793"},"reference-count":38,"publisher":"The Royal Society","issue":"2328","license":[{"start":{"date-parts":[[2026,8,28]],"date-time":"2026-08-28T00:00:00Z","timestamp":1787875200000},"content-version":"vor","delay-in-days":0,"URL":"http:\/\/creativecommons.org\/licenses\/by\/4.0\/"},{"start":{"date-parts":[[2026,8,28]],"date-time":"2026-08-28T00:00:00Z","timestamp":1787875200000},"content-version":"vor","delay-in-days":0,"URL":"http:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100001711","name":"Schweizerischer Nationalfonds zur F\u00f6rderung der Wissenschaftlichen Forschung","doi-asserted-by":"crossref","award":["project number 223790"],"award-info":[{"award-number":["project number 223790"]}],"id":[{"id":"10.13039\/501100001711","id-type":"DOI","asserted-by":"crossref"}]},{"DOI":"10.13039\/501100016047","name":"Science Fund of the Republic of Serbia","doi-asserted-by":"publisher","award":["grant number 7749891"],"award-info":[{"award-number":["grant number 7749891"]}],"id":[{"id":"10.13039\/501100016047","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2026,8,28]]},"abstract":"<jats:title>Abstract<\/jats:title>\n                  <jats:p>We establish compatibility of Lie structures that appear in homotopy calculus of functors and isotopy calculus of embeddings. On one hand, we give a new proof of the Johnson\u2013Arone\u2013Mahowald result describing the layers of the Goodwillie tower of the identity functor, and we directly compare the spectral Lie bracket with the classical Samelson (Whitehead) bracket on spaces. On the other hand, we geometrically define a bracket on the layers of the embedding calculus tower for embeddings of arcs. These results are unified through the same technical tool, a newly defined bracket on total homotopy fibres of collapsing cubes of wedge sums.<\/jats:p>\n                  <jats:p>This article is part of the theme issue \u2018Derived Lie algebras in geometry and topology\u2019.<\/jats:p>","DOI":"10.1098\/rsta.2024.0403","type":"journal-article","created":{"date-parts":[[2026,8,27]],"date-time":"2026-08-27T08:06:36Z","timestamp":1787817996000},"source":"Crossref","is-referenced-by-count":0,"title":["Lie structures in homotopy and isotopy calculi"],"prefix":"10.1098","volume":"384","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-3923-4587","authenticated-orcid":false,"given":"Danica","family":"Kosanovi\u0107","sequence":"first","affiliation":[{"name":"University of Bern, 1","place":["Bern,, Switzerland"],"department":["Mathematisches Institut (MAI),"]},{"name":"CNRS, IMJ-PRG, Universit\u00e9 Paris Cit\u00e9, Sorbonne Universit\u00e9, 2","place":["Paris,, France"]}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"175","published-online":{"date-parts":[[2026,8,28]]},"reference":[{"key":"2026082804090316600_R1","doi-asserted-by":"crossref","first-page":"645","DOI":"10.2140\/gt.2003.7.645","article-title":"Calculus III: Taylor series","volume":"7","author":"Goodwillie","year":"2003","journal-title":"Geom. Topol."},{"key":"2026082804090316600_R2","doi-asserted-by":"crossref","first-page":"67","DOI":"10.2140\/gt.1999.3.67","article-title":"Embeddings from the point of view of immersion theory. I","volume":"3","author":"Weiss","year":"1999","journal-title":"Geom. Topol."},{"key":"2026082804090316600_R3","doi-asserted-by":"crossref","first-page":"103","DOI":"10.2140\/gt.1999.3.103","article-title":"Embeddings from the point of view of immersion theory. II","volume":"3","author":"Goodwillie","year":"1999","journal-title":"Geom. Topol."},{"key":"2026082804090316600_R4","doi-asserted-by":"crossref","first-page":"651","DOI":"10.1112\/jtopol\/jtv008","article-title":"Multiple disjunction for spaces of smooth embeddings","volume":"8","author":"Goodwillie","year":"2015","journal-title":"J. Topol."},{"key":"2026082804090316600_R5","first-page":"221","volume-title":"Surveys on Surgery Theory. Annals of Mathematics Studies","author":"Goodwillie","year":"2001"},{"key":"2026082804090316600_R6","doi-asserted-by":"crossref","first-page":"1161","DOI":"10.2140\/agt.2013.13.1161","article-title":"Cosimplicial models for the limit of the Goodwillie tower","volume":"13","author":"Eldred","year":"2013","journal-title":"Algebr. Geom. Topol."},{"key":"2026082804090316600_R7","first-page":"1","volume-title":"Stable and unstable homotopy","author":"Arone","year":"1998"},{"key":"2026082804090316600_R8","doi-asserted-by":"crossref","first-page":"33","DOI":"10.2140\/agt.2010.10.33","article-title":"Homotopy nilpotent groups","volume":"10","author":"Biedermann","year":"2010","journal-title":"Algebr. Geom. Topol."},{"key":"2026082804090316600_R9","doi-asserted-by":"crossref","first-page":"119","DOI":"10.1515\/forum-2012-0038","article-title":"Goodwillie calculus and Whitehead products","volume":"27","author":"Chorny","year":"2015","journal-title":"Forum Math."},{"key":"2026082804090316600_R10","doi-asserted-by":"crossref","first-page":"1295","DOI":"10.1090\/S0002-9947-1995-1297532-6","article-title":"The derivatives of homotopy theory","volume":"347","author":"Johnson","year":"1995","journal-title":"Trans. Am. Math. Soc."},{"key":"2026082804090316600_R11","doi-asserted-by":"crossref","first-page":"833","DOI":"10.2140\/gt.2005.9.833","article-title":"Bar constructions for topological operads and the Goodwillie derivatives of the identity","volume":"9","author":"Ching","year":"2005","journal-title":"Geom. Topol."},{"key":"2026082804090316600_R12","doi-asserted-by":"crossref","first-page":"223","DOI":"10.4007\/annals.2021.193.1.3","article-title":"Lie algebras and vn-periodic spaces","volume":"193","author":"Heuts","year":"2021","journal-title":"Ann. Math."},{"key":"2026082804090316600_R13","doi-asserted-by":"crossref","first-page":"167","DOI":"10.4310\/HHA.2020.v22.n1.a10","article-title":"The vn-periodic Goodwillie tower on wedges and cofibres","volume":"22","author":"Brantner","year":"2020","journal-title":"Homol. Homotopy Appl."},{"key":"2026082804090316600_R14","doi-asserted-by":"crossref","first-page":"361","DOI":"10.4310\/HHA.2013.v15.n2.a20","article-title":"Manifold calculus and homotopy sheaves","volume":"15","author":"Boavida de Brito","year":"2013","journal-title":"Homol. Homotopy Appl."},{"key":"2026082804090316600_R15","article-title":"A context for manifold calculus","author":"Arakawa","year":"2024","journal-title":"arXiv: 2403.03321."},{"key":"2026082804090316600_R16","doi-asserted-by":"crossref","DOI":"10.1016\/j.aim.2024.109779","article-title":"Embedding calculus and grope cobordism of knots","volume":"451","author":"Kosanovi\u0107","year":"2024","journal-title":"Adv. Math."},{"key":"2026082804090316600_R17","doi-asserted-by":"crossref","first-page":"e12337","DOI":"10.1112\/topo.12337","article-title":"Knotted families from graspers","volume":"17","author":"Kosanovi\u0107","year":"2024","journal-title":"J. Topol."},{"key":"2026082804090316600_R18","doi-asserted-by":"crossref","first-page":"799","DOI":"10.4171\/cmh\/578","article-title":"A space level light bulb theorem in all dimensions","volume":"99","author":"Kosanovi\u0107","year":"2024","journal-title":"Math. Helvetici"},{"key":"2026082804090316600_R19","doi-asserted-by":"crossref","first-page":"743","DOI":"10.1007\/s002220050300","article-title":"The Goodwillie tower of the identity functor and the unstable periodic homotopy of spheres","volume":"135","author":"Arone","year":"1999","journal-title":"Invent. Math."},{"key":"2026082804090316600_R20","doi-asserted-by":"crossref","DOI":"10.1112\/topo.12334","article-title":"Koszul self-duality of manifolds","volume":"17","author":"Malin","year":"2024","journal-title":"J. Topol."},{"key":"2026082804090316600_R21","unstructured":"Ching\n              M\n            \n          . Calculus of functors and configuration spaces. InConf. Pure and Applied Topology Isle of Skye. https:\/\/mching.people.amherst.edu\/Work\/skye.pdf."},{"key":"2026082804090316600_R22","doi-asserted-by":"crossref","first-page":"383","DOI":"10.1112\/jtopol\/jtr005","article-title":"Derivatives of the identity and generalizations of Milnor\u2019s invariants","volume":"4","author":"Munson","year":"2011","journal-title":"J. Topol."},{"key":"2026082804090316600_R23","doi-asserted-by":"crossref","first-page":"1079","DOI":"10.2140\/agt.2019.19.1079","article-title":"Homotopical intersection theory, III: multirelative intersection problems","volume":"19","author":"Klein","year":"2019","journal-title":"Algebr. Geom. Topol."},{"key":"2026082804090316600_R24","doi-asserted-by":"crossref","first-page":"813","DOI":"10.1016\/0040-9383(92)90012-7","article-title":"A spectrum level rank filtration in algebraic K-theory","volume":"31","author":"Rognes","year":"1992","journal-title":"Topology"},{"key":"2026082804090316600_R25","doi-asserted-by":"crossref","unstructured":"Fresse\n              B\n            \n          . 2004Koszul duality of operads and homology of partition posets. InHomotopy Theory: Relations With Algebraic Geometry, Group Cohomology, and Algebraic K-Theory. Papers From the Int. Conf. Algebraic Topology, Northwestern University, Evanston, IL, pp. 115\u2013215. Providence, RI: American Mathematical Society.","DOI":"10.1090\/conm\/346\/06287"},{"key":"2026082804090316600_R26","doi-asserted-by":"crossref","first-page":"699","DOI":"10.1016\/j.jpaa.2006.03.012","article-title":"Homology of generalized partition posets","volume":"208","author":"Vallette","year":"2007","journal-title":"J. Pure Appl. Algebra"},{"key":"2026082804090316600_R27","doi-asserted-by":"crossref","first-page":"47","DOI":"10.1007\/s10240-021-00123-7","article-title":"The action of young subgroups on the partition complex","volume":"133","author":"Arone","year":"2021","journal-title":"Publ. Math. Inst. Hautes \u00c9tudes Sci."},{"key":"2026082804090316600_R28","doi-asserted-by":"crossref","first-page":"341","DOI":"10.1353\/ajm.2008.0020","article-title":"Homotopy approximations to the space of knots, Feynman diagrams, and a conjecture of Scannell and Sinha","volume":"130","author":"Conant","year":"2008","journal-title":"Am. J. Math."},{"key":"2026082804090316600_R29","doi-asserted-by":"crossref","first-page":"943","DOI":"10.2140\/agt.2004.4.943","article-title":"Partition complexes, duality and integral tree representations","volume":"4","author":"Robinson","year":"2004","journal-title":"Algebr. Geom. Topol."},{"key":"2026082804090316600_R30","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/S0022-4049(97)00050-9","article-title":"The homology of certain subgroups of the symmetric group with coefficients in Lie(n)","volume":"127","author":"Arone","year":"1998","journal-title":"J. Pure Appl. Algebr."},{"key":"2026082804090316600_R31","doi-asserted-by":"crossref","first-page":"93","DOI":"10.1016\/0097-3165(90)90050-7","article-title":"On the action of the symmetric group on the Free Lie Algebra and the partition lattice","volume":"55","author":"Barcelo","year":"1990","journal-title":"J. Comb. Theory Ser. A"},{"key":"2026082804090316600_R32","doi-asserted-by":"crossref","first-page":"287","DOI":"10.1016\/S0012-365X(98)00147-2","article-title":"On the (co)homology of the partition lattice and the free Lie algebra","volume":"193","author":"Wachs","year":"1998","journal-title":"Discrete Math."},{"key":"2026082804090316600_R33","doi-asserted-by":"crossref","DOI":"10.1007\/978-1-4612-6318-0","volume-title":"Elements of homotopy theory","author":"Whitehead","year":"1978"},{"key":"2026082804090316600_R34","volume-title":"New mathematical monographs","author":"Munson","year":"2015"},{"key":"2026082804090316600_R35","doi-asserted-by":"crossref","first-page":"199","DOI":"10.1016\/0040-9383(71)90004-8","article-title":"A note on the Hilton\u2013Milnor theorem","volume":"10","author":"Gray","year":"1971","journal-title":"Topology"},{"key":"2026082804090316600_R36","first-page":"569","article-title":"Variations of knotted graphs. The geometric technique of n-equivalence","volume":"12","author":"Gusarov","year":"2000","journal-title":"Algebra i Analiz"},{"key":"2026082804090316600_R37","doi-asserted-by":"crossref","first-page":"1","DOI":"10.2140\/gt.2000.4.1","article-title":"Claspers and finite type invariants of links","volume":"4","author":"Habiro","year":"2000","journal-title":"Geom. Topol."},{"key":"2026082804090316600_R38","doi-asserted-by":"crossref","first-page":"119","DOI":"10.1016\/S0040-9383(03)00031-4","article-title":"Grope cobordism of classical knots","volume":"43","author":"Conant","year":"2004","journal-title":"Topology"}],"container-title":["Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/royalsocietypublishing.org\/rsta\/article-pdf\/doi\/10.1098\/rsta.2024.0403\/6163212\/rsta.2024.0403.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"syndication"},{"URL":"https:\/\/royalsocietypublishing.org\/rsta\/article-pdf\/doi\/10.1098\/rsta.2024.0403\/6163212\/rsta.2024.0403.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,28]],"date-time":"2026-08-28T08:09:12Z","timestamp":1787904552000},"score":1,"resource":{"primary":{"URL":"https:\/\/royalsocietypublishing.org\/rsta\/article\/384\/2328\/20240403\/483056\/Lie-structures-in-homotopy-and-isotopy-calculi"}},"subtitle":[],"editor":[{"given":"Lukas","family":"Brantner","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"editor"}]}],"short-title":[],"issued":{"date-parts":[[2026,8,28]]},"references-count":38,"journal-issue":{"issue":"2328","published-online":{"date-parts":[[2026,8,28]]},"published-print":{"date-parts":[[2026,8,28]]}},"URL":"https:\/\/doi.org\/10.1098\/rsta.2024.0403","relation":{},"ISSN":["1364-503X","1471-2962"],"issn-type":[{"value":"1364-503X","type":"print"},{"value":"1471-2962","type":"electronic"}],"subject":[],"published":{"date-parts":[[2026,8,28]]},"article-number":"20240403"}}