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On Fundamental Groups of Galois Closures of Generic Projections

On Fundamental Groups of Galois Closures of Generic Projections

Examples 5.21 The

Examples 5.21 The following H 2 ’s vanishH 2 (¡ n) = H 2 (Q 8 ) = H 2 (¡ ) = H 2 (D ∞ ) = 1,where Q 8 denotes the quaternion group and D ∞ denotes the infinite dihedralgroup. For the dihedral groups of order 2n we have{1 if n is oddH 2 (D 2n ) =2 if n ¡ is even.For n ≥ 4 it is known thatAppendix to Section 5H 2 (¡ 2 × ¡ 2) = H 2 (S n ) = ¡ 2.5.5 Group homology and the computation of H 2In this section we first recall the construction of group homology. Then we givesome of its properties and give some statements that allow us to actually computeH 2 of a given group. As references we refer to [Br, Chapter II], [We, Chapter 6],[Rot, Chapter 7] and [Rot, Chapter 11].Let G be an arbitrary group. For a left G-module M we define its module ofco-invariants to be the quotient of M by the module I G generated by all elementsg · m − m for all g ∈ G and m ∈ M:M G := M/I G .Taking co-invariants defines a right exact functor for left G-modules and we canconsider its left derived functor. We define the i.th homology H i (G) of G to be thei.th left derived functor of − G applied to the G-module ¡ with trivial G-action:H i (G) := H i (G, ¡ ).Using the standard resolution of ¡ over the group ring ¡ [G] it is not hard to provethat for all groupsH 0 (G) ∼ = ¡H 1 (G) ∼ = Gabholds true. Clearly, all homology groups are abelian groups. Using again thestandard resolution mentioned before one can show that if G is a finite group thenalso its homology groups are finite.The origins of group homology lie in algebraic topology: We recall that aconnected CW-complex Y is called a K(G, 1)-complex if π top1 (Y ) ∼ = G and if itsuniversal cover is contractible.58

Theorem 5.22 For a K(G, 1)-complex Y there exist for all i ≥ 0 isomorphismsH i (G) ∼ = Hi (Y, ¡ )where H i (Y, ¡ ) denotes the singular homology of the topological space Y .A short exact sequence0 → A → X → G → 1is called a central extension of G if A lies in the centre of X. A central extension0 → A → X → G → 1 is called a universal central extension if for every centralextension 1 → B → Y → G → 1 there exists a unique homomorphism from Xto Y making the following diagram commute0 → A → X → G → 1↓ ↓ ||0 → B → Y → G → 1If such a universal central extension exists it is unique up to isomorphism.Central extensions 0 → A → X → G → 1 with a fixed abelian group Aare classified by Hom(H 2 (G), A). In particular, central extensions with ¨ ∗ areclassified by Hom(H 2 (G), ¨ ∗ ) ∼ = H 2 (G, ¨ ∗ ) =: M(G). This latter group iscalled the Schur multiplier of G. If G is finite then Pontryagin duality provides uswith a non-canonical isomorphism between H 2 (G) and M(G).A group G has a universal central extension ˜G if and only if it is perfect. Inthis case the universal extension takes the form0 → H 2 (G) → ˜G → G → 1.Now let N be a normal subgroup of a free group F such that G ∼ = F/N. Thenthere is a central extension0 → (N ∩ [F, F ])/[N, F ] → [F, F ]/[N, F ] → [G, G] → 1.In case G is a perfect group this is exactly its universal central extension. But evenin the case where G is not necessarily perfect we have the followingTheorem 5.23 (Hopf) Let G be an arbitrary group. If N is a normal subgroupof a free group F such that G ∼ = F/N thenH 2 (G) ∼ = (N ∩ [F, F ])/[F, N].59

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