Two extensions of the essential annulus theorem

✍ Scribed by C.D. Feustel

• Publisher: Elsevier Science
• Year: 1977
• Weight: 840 KB
• Volume: 7
• Category: Article
• ISSN: 0016-660X
• DOI: 10.1016/0016-660x(77)90028-9

✦ Synopsis

Let M be a 3 rulifold, A an annulus, and (2 a spanning arc of A, i.e. A --01 is connect&d' and simply connected. A map f : (A, aA)-+ (M, HIYQ is essent,ial if j$ : ?r@)-=+ wi(M)'i's monk and j'(a) is not ho:motopic rei its boundary to an ax in 8Ak W~Z say that f is a D&n map if p /&4 is an embedding. Ilw this note we show that 112 c&&n 6ort & essentiai map can be replaced by th t same sort of Dehn map This is especia'lly imfiortant in view of [ 1, Theoxems 1 and 1') where it is &own th';tt an essential Dehn map of an annufus can be replaced by an Dehn embedding,.

We remark thal the reiation of Theorem 1 to the essential annulus theorem iis roughly the sz;z.: as that ,of the Loop Thesrem f4,Sj to Del:n's Lemma Th&author wi&es to thank the referee for pointing ou: ail error in the oti proof.' W$ &note the comporeents of dA by ct. antd ~2 and the wit internal by 1, define i spanning, arc and an essential map" sf a M&us &nif&ds are simpli&l complexes and all makt are piece I %'hii?i~~ap k Let M 'be a compact, orientable, ime&cibk 3 manrfakf and &$om_~~e&ble surfade in aM. Let f : (A, i)A ) -+ (M, F) be an c": sentiutb map. Le CL kotiat &bgmup of n,(M). If [f(ct)] a-f A?, there exi:vts an ~~senf~zii g : (A, dA ) -(M, F) such that 1) jj 1 dA is aft em bedding. 2). [g(G)] & \N 3) i (cu ) is homotopic rel its boundary to the pmduct off (cty ) cm-i tw 3 Tl~earern 2. Let .M be a compuct, orien~cabk4, imduc i;lrc&p~e&ble suvface in t9M. Let f : (A, 8A )-+ (M, F) * The author is partially suppwted by NSF Cirant GP' 15357.