<\/p>\n\n\n\n
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The following problem has been posed by G. E. Bredon in his book “Introduction to compact transformation groups<\/em>”, Academic Press, 1972, page 58.<\/p>\n\n\n\n Bredon’s Problem.<\/strong> If a compact Lie group G<\/em> acts on a disk, Euclidean space, or sphere, then is it true that the fixed point set always<\/em> consists of connected components of the same dimension?<\/p>\n\n\n\n For smooth actions, the answer to the question above is given in my paper “Group actions with inequivalent representations at fixed points<\/em>”, Math. Zeitschrift, Vol. 187 (1984), pages 29–47. The answer depends on the quotient group G<\/em>\/G* <\/em>of G<\/em>, where G*<\/em> is the connected component of the identity element of G<\/em>.<\/p>\n\n\n\n Theorem.<\/strong> Let G be a compact Lie group<\/em>. Then for any smooth action of G on an acyclic manifold (resp., a homology sphere), each connected component of the fixed point set has the same dimension if and only if G\/G* does not contain an element of prime power order<\/em>, i.e.<\/em>, G is connected or each element of G\/G* is of prime power order<\/em>.<\/p>\n\n\n\n The sufficiency of the restriction on G<\/em> follows from the fact that if G<\/em> is connected or each element of G\/G*<\/em> is of prime power order, then for any smooth action of G<\/em> on an acyclic manifold (resp., a homology sphere with at least three fixed points), the representations of G<\/em> on the tangent spaces at any two fixed points, determined via the derivation of group action, are isomorphic to each other and therefore, by the Slice Theorem, the dimension conclusion holds.<\/p>\n\n\n\n The necessity of the restriction on G<\/em> is shown by proving that if G\/G*<\/em> contains an element not of prime power order, then there exists a smooth action of G<\/em> on a disk, resp., Euclidean space; sphere, such that the fixed point set contains connected components of distinct dimensions.<\/p>\n\n\n\n <\/p>\n","protected":false},"excerpt":{"rendered":" The following problem has been posed by G. E. Bredon in his book “Introduction to compact transformation groups”, Academic Press, 1972, page 58. Bredon’s Problem. If a compact Lie group G acts on a disk, Euclidean space, or sphere, then is it true that the fixed point set always consists ( more… )<\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"parent":16,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"kt_blocks_editor_width":""},"_links":{"self":[{"href":"https:\/\/kpa.faculty.wmi.amu.edu.pl\/index.php?rest_route=\/wp\/v2\/pages\/20"}],"collection":[{"href":"https:\/\/kpa.faculty.wmi.amu.edu.pl\/index.php?rest_route=\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/kpa.faculty.wmi.amu.edu.pl\/index.php?rest_route=\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/kpa.faculty.wmi.amu.edu.pl\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/kpa.faculty.wmi.amu.edu.pl\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=20"}],"version-history":[{"count":157,"href":"https:\/\/kpa.faculty.wmi.amu.edu.pl\/index.php?rest_route=\/wp\/v2\/pages\/20\/revisions"}],"predecessor-version":[{"id":2030,"href":"https:\/\/kpa.faculty.wmi.amu.edu.pl\/index.php?rest_route=\/wp\/v2\/pages\/20\/revisions\/2030"}],"up":[{"embeddable":true,"href":"https:\/\/kpa.faculty.wmi.amu.edu.pl\/index.php?rest_route=\/wp\/v2\/pages\/16"}],"wp:attachment":[{"href":"https:\/\/kpa.faculty.wmi.amu.edu.pl\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=20"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}