{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2022,4,1]],"date-time":"2022-04-01T18:29:47Z","timestamp":1648837787991},"reference-count":15,"publisher":"World Scientific Pub Co Pte Lt","issue":"05","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Int. J. Algebra Comput."],"published-print":{"date-parts":[[2008,8]]},"abstract":"<jats:p> We show that the intersection of three subgroups in a free group is related to the computation of the third homotopy group \u03c0<jats:sub>3<\/jats:sub>. This generalizes a result of Gutierrez\u2013Ratcliffe who relate the intersection of two subgroups with the computation of \u03c0<jats:sub>2<\/jats:sub>. Let K be a two-dimensional CW-complex with subcomplexes K<jats:sub>1<\/jats:sub>, K<jats:sub>2<\/jats:sub>, K<jats:sub>3<\/jats:sub> such that K = K<jats:sub>1<\/jats:sub> \u222a K<jats:sub>2<\/jats:sub> \u222a K<jats:sub>3<\/jats:sub> and K<jats:sub>1<\/jats:sub> \u2229 K<jats:sub>2<\/jats:sub> \u2229 K<jats:sub>3<\/jats:sub> is the 1-skeleton K<jats:sup>1<\/jats:sup> of K. We construct a natural homomorphism of \u03c0<jats:sub>1<\/jats:sub>(K)-modules [Formula: see text] where R<jats:sub>i<\/jats:sub> = ker {\u03c0<jats:sub>1<\/jats:sub>(K<jats:sup>1<\/jats:sup>) \u2192 \u03c0<jats:sub>1<\/jats:sub>(K<jats:sub>i<\/jats:sub>)}, i = 1,2,3 and the action of \u03c0<jats:sub>1<\/jats:sub>(K) = F\/R<jats:sub>1<\/jats:sub>R<jats:sub>2<\/jats:sub>R<jats:sub>3<\/jats:sub> on the right-hand abelian group is defined via conjugation in F. In certain cases, the defined map is an isomorphism. Finally, we discuss certain applications of the above map to group homology. <\/jats:p>","DOI":"10.1142\/s0218196708004652","type":"journal-article","created":{"date-parts":[[2008,8,25]],"date-time":"2008-08-25T09:25:17Z","timestamp":1219656317000},"page":"803-823","source":"Crossref","is-referenced-by-count":1,"title":["INTERSECTION OF SUBGROUPS IN FREE GROUPS AND HOMOTOPY GROUPS"],"prefix":"10.1142","volume":"18","author":[{"given":"HANS-JOACHIM","family":"BAUES","sequence":"first","affiliation":[{"name":"Max-Planck-Institut fur Mathematik, Vivatsgasse 7, 53111, Bonn, Germany"}]},{"given":"ROMAN","family":"MIKHAILOV","sequence":"additional","affiliation":[{"name":"Steklov Mathematical Institute, Gubkina 8, 119991 Moscow, Russia"}]}],"member":"219","published-online":{"date-parts":[[2011,11,20]]},"reference":[{"key":"rf1","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9781107325890"},{"key":"rf2","doi-asserted-by":"publisher","DOI":"10.1016\/0022-4049(92)90030-J"},{"key":"rf3","doi-asserted-by":"publisher","DOI":"10.1016\/0040-9383(66)90024-3"},{"key":"rf4","doi-asserted-by":"publisher","DOI":"10.1112\/blms\/20.2.124"},{"key":"rf5","doi-asserted-by":"publisher","DOI":"10.1016\/0001-8708(71)90015-6"},{"key":"rf6","doi-asserted-by":"publisher","DOI":"10.1007\/s10977-005-3115-5"},{"key":"rf7","doi-asserted-by":"publisher","DOI":"10.1017\/S0017089500031062"},{"key":"rf8","doi-asserted-by":"publisher","DOI":"10.1007\/BF02566679"},{"key":"rf9","doi-asserted-by":"publisher","DOI":"10.1016\/0022-4049(86)90139-8"},{"key":"rf10","doi-asserted-by":"publisher","DOI":"10.1093\/qmath\/32.1.45"},{"key":"rf12","doi-asserted-by":"publisher","DOI":"10.1016\/0022-4049(91)90115-I"},{"key":"rf13","first-page":"111","volume":"9","author":"Multu A.","journal-title":"Appl. Cat. Str."},{"key":"rf14","first-page":"687","author":"Pride S. J.","journal-title":"Proc. Group Theory from a Geometrical Point of View"},{"key":"rf15","series-title":"Annals of Mathematics Studies, No. 49","volume-title":"Composition methods in homotopy groups of spheres","author":"Toda H.","year":"1962"},{"key":"rf16","doi-asserted-by":"crossref","first-page":"489","DOI":"10.1017\/S030500410100487X","volume":"130","author":"Wu J.","journal-title":"Math. Proc. Camb. Phyl. Soc."}],"container-title":["International Journal of Algebra and Computation"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.worldscientific.com\/doi\/pdf\/10.1142\/S0218196708004652","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,8,6]],"date-time":"2019-08-06T22:28:16Z","timestamp":1565130496000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.worldscientific.com\/doi\/abs\/10.1142\/S0218196708004652"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2008,8]]},"references-count":15,"journal-issue":{"issue":"05","published-online":{"date-parts":[[2011,11,20]]},"published-print":{"date-parts":[[2008,8]]}},"alternative-id":["10.1142\/S0218196708004652"],"URL":"https:\/\/doi.org\/10.1142\/s0218196708004652","relation":{},"ISSN":["0218-1967","1793-6500"],"issn-type":[{"value":"0218-1967","type":"print"},{"value":"1793-6500","type":"electronic"}],"subject":[],"published":{"date-parts":[[2008,8]]}}}