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|2.1.4 A Proof of the Dehn-Nielsen Theorem Let us begin with the following |
definition. Suppose that X and Y are 2- orbifolds and that f:X — > Y is a
continuous map between the underlying topological spaces. We say that f is an
orbifold map if ...
|by Fenchel and an English translation was published in Nielsen's Collected |
Mathematical Papers (1986). The notes credit the theorem to Dehn and say that
Dehn's proof uses the idea from Poincaré (1904) of lifting curves to the universal
|We have the following remarkable theorem. THEOREM 8.1 (Dehn–Nielsen–Baer|
) Let g ≥ 1. The homomorphism σ : Mod±(Sg) −→ Out(π1(Sg)) is an isomorphism
. As noted above, the proof of Theorem 8.1 reduces to the statement that σ is ...
|It is faithful, more precisely: Theorem 2.19 (Artin , ). (1) The Artin ... The first |
way uses the Fadell-Neuwirth fiber bundle p: Mn+\ -»□ M„ of Theorem 2.10. Let
Sym„ act ... In this setting, it is known as the Dehn-Nielsen-Baer theorem. Here is
|For closed S, the surjectivity part of these theorems is due to Dehn, who did not |
publish his proof, and to Nielsen , who published a proof partially based on
Dehn's ideas. Another proof was suggested by Seifert . The injectivity part is
|Dehn-Nielsen Theorem. An automorphism a of the fundamental group G of a |
compact surface is induced by a homeomorphism of the surface if and only if Q(si
) = wis^w^l . 1 < i fc < m. In this case (^ '",!") is a permutation and wi C G, w(u>i)ei
|Since both result from systems of cuts of the surface S and these systems are of |
the same type it follows that there is a homeomorphism inducing a: 3.4.6. Dehn-
Nielsen Theorem. An automorphism a of the fundamental group G of a compact ...
|In this case, by Theorem 1.4 (or Theorem 2.14) below, Modg,p and Mod+g,p are |
not isomorphic to Out(π1(Sg,p)), and hence the exact analogue of Dehn-Nielsen
theorem does not hold. On the other hand, a modification holds by requiring that
|is an inner automorphism of tti(M), and so [ft*] = 1. Thus, the mapping h t— > [ft*] |
induces a function v : Map(M) -> Out(?ri(M)). The next theorem is a basic theorem
of surface theory. Theorem 9.4.3. (The Dehn-Nielsen Theorem) If M is a closed ...
|width of 70 collar theorems 94,97, 106, 112 color 298 commensurable 336 |
complete sequence 379 completely ... 111 cut open 15 cycle 313 defining
relation 164, 171, 179 D Dehn twist 72,425 Dehn-Nielsen theorem 171 direct 156