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By Tammo tom Dieck

This e-book is a jewel– it explains vital, precious and deep issues in Algebraic Topology that you simply won`t locate in other places, rigorously and in detail."""" Prof. Günter M. Ziegler, TU Berlin

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Extra resources for Algebraic Topology and Transformation Groups

Example text

S The space HL1 (GF,S , adρ¯)(η) is readily identified with the tangent space of (η) (η) RS,O . For its dimension we write h . Thus we have presentations: (η) 0 −→ J (η) −→ O[[T1 , . . , Th(η) ]] −→ RS,O −→ 0. 2) Note that Im(H 0 (GF,S , adρ¯/ad0ρ¯) → H 1 (GF,S , ad0ρ¯)) injects under the canonical restriction homomorphism into each of the H 1 (Gν , ad0ρ¯). From this and our definition of the L0ν , one deduces that there is a short exact sequence 0 → Im(H 0 (GF,S , adρ¯/ad0ρ¯) → H 1 (GF,S , ad0ρ¯)) → HL1 0 (GF,S , ad0ρ¯) → HL1 (GF,S , adρ¯)η → 0.

We define δν,unr to be 1 in case (i) and 0 in case (ii). Then we have h0ν = dimF tRη ν,O = 1 + h1 (Gν , χ ¯2 η¯ν−1 ) − δν,unr . Similarly, one can compute the obstruction to further lift a representation ρ : Gν → GL2 (R) : σ → to a representation ρ given by χ 0 b (χ )−1 ην χ b 0 χ−1 ην for a small surjection R → R. Letting χ be an unramified character which lifts χ (and always exists since ˆ is of cohomological dimension one) and b a set-theoretic Gν /Iν ∼ = Z continuous lift, as is standard, one shows that (s, t) → ρ (st)ρ (t)−1 ρ (s)−1 =: 1 cs,t 0 1 defines a 2-cocycle of Gν with values in χ ¯2 η¯ν−1 , and so we obtain a class in η 2 2 −1 H (Gν , χ ¯ η¯ν ).

1) for the functors Def ν,O and Def ν,O , respectively: / O[[Tν,1, . . , T (η) ]] ν,h / / R(η) ν,O (η) Jν / O[[Tν,1, . . , T (η) ]] ν,h //   (η) Jν  _  (η) Jν (η) The ideal Jν (η) ν ν  (η) Rν,O πν / O[[Tν,1, . . , T (η) ]] ν,h ν / / R(η) . ν,O (η) is the kernel of the composite O[[Tν,1, . . , Tν,h(η) ]] → Rν,O → ν Rν,O . The epimorphism πν is chosen so that the lower right square commutes. We may rearrange the variables in such a way that πν is concretely (η) given by mapping Tν,i to Tν,i , for i ≤ hν , and by mapping Tν,i to zero (η) for i > hν .

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