By Alejandro Adem, José Manuel Gómez (auth.), A. Bjorner, F. Cohen, C. De Concini, C. Procesi, M. Salvetti (eds.)

These lawsuits comprise the contributions of a few of the individuals within the "intensive study interval" held on the De Giorgi learn heart in Pisa, in the course of the interval May-June 2010. The vital topic of this learn interval used to be the research of configuration areas from a number of issues of view. This subject originated from the intersection of numerous classical theories: Braid teams and similar issues, configurations of vectors (of nice value in Lie concept and illustration theory), preparations of hyperplanes and of subspaces, combinatorics, singularity thought. lately, in spite of the fact that, configuration areas have obtained autonomous curiosity and certainly the contributions during this quantity move a long way past the above matters, making it appealing to a wide viewers of mathematicians.

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**Example text**

We can choose each f i in such a way that the isotropy subgroup W fi is a reﬂection subgroup of W of the form W Ii for some Ii ⊂ . For each 1 ≤ i ≤ m let W fi be a system of minimal length representatives of W/W fi as deﬁned above. Deﬁne m u −1 K ( ) × { f i } ⊂ T × Hom(A, T ) = X T . L( ) := i=1 u∈W fi Deﬁned in this way L( ) ⊂ X T is a sub CW-complex. We now show that L( ) is such that for every element x ∈ X T there is a unique w ∈ W such that wx ∈ L( ). To see this, since K ( ) ⊂ T satisﬁes this property, it sufﬁces to see that for any i and any v1 , v2 ∈ W there are unique v ∈ W and u ∈ W fi such that v1 K ( ) × v2 f i = v(u −1 K ( ) × f i ).

The same argument shows that the inclusion map i : G/T ×W F(π, f ) → G/T ×W Hom(π, G)1T1 f 19 On the structure of spaces of commuting elements in compact Lie groups induces a surjective homomorphism i ∗ : π1 (G/T ×W F(π, f )) → π1 (G/T ×W Hom(π, G)1T1 f ). 6]). 3. Suppose that G ∈ P and that π is a ﬁnitely generated abelian group. Then the map ϕ : G/T ×W Hom(π, G)1T1 f → Hom(π, G)11 f is π1 -surjective. Note that G/T ×W Hom(π, G)1T1 f ∼ = G/T ×W f T n . Since W f acts freely on G/T the projection map p induces a ﬁbration sequence p T n → G/T ×W f T n → (G/T )/W f ∼ = G/NG f (T ).

The tools applied in [7] can be 17 On the structure of spaces of commuting elements in compact Lie groups used to generalize this result to the class of spaces of homomorphisms Hom(π, G). Here we need to assume that π is a ﬁnitely generated abelian group and that G is a Lie group in the class P. 1] can be generalized for any choice of base point in Hom(π, G). Write π in the form π = Zn ⊕ A, where A is a ﬁnite abelian group. Suppose ﬁrst that n = 0 and thus π is a ﬁnite group. 5 there is a homeomorphism G/G f , : Hom(π, G) → [ f ]∈Hom(π,T )/W where each G f is a maximal rank isotropy subgroup with W (G f ) = W f .