By Zoque E.
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Additional resources for A basis for the non-crossing partition lattice top homology
For a finite group G there is a standard resolution: i0 i1 i2 1 ← Z ←− Z[G] ←− Z[G]#G ←− Z[G]... ← . . with the boundary homomorphism ∂r [g 1 , . . , g r ] = g 1 [g 2 , . . , g r ] + (−1)i [g 1 , . . , (g i g i +1 ), . . g r ] +(−1)r [g 1 , . . , g r ] This complex is isomorphic to the simplicial chain complex of a simplicial space BG with r -simplices indexed by [g 1 , . . , g r ]. The space BG is in fact a quotient EG/G of another space EG represented by a simplicial complex with a free action of G.
For an arbitrary G, the complex EG contains complexes E g , g ∈ G isomorphic to E e and we can contract all of them independently E g → g ∈ EG where g is a 0-dimensional subcomplex G ⊂ EG. Now we can define a contraction of EG to the simplex ∆ = (g 1 , . . , g n ), where g i runs through G once without repetitions. Any simplex (g 1 , . . , g N ) (now g i may repeat) will contract to a subsimplex of ∆ which is a linear combination of different vertices of (g 1 , . . , g N ). Thus, we have natural F.
In the second case the image f ∗ (I νD ) is contained in I ν f ∗ (D) and f ∗ (G al νD ) ⊂ G al ν f ∗ (D) , so again f ∗ a is unramified. Assume now that X ⊂ Y is an embedding of irreducible varieties over k and assume that generic point of X is smooth in Y . Using induction we can assume that X is a divisor on Y corresponding to valuation ν X . Then the decomposition group G al ν (k(Y )) = G al (k(X )) × I ν where I ν is a topologically cyclic group. Thus any ∗ element a ∈ Hnr (G al (k(Y ), Z/p) restricts to an element of H ∗ (G al (k(X ), Z/p) ⊂ ∗ ∗ (G al (k(Y )) → H ∗ (G al (k(X ), Z/p) is H (G al ν (k(Y )) and hence the map f ∗ : Hnr well-defined.
A basis for the non-crossing partition lattice top homology by Zoque E.