By Loren Bliese
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Additional info for A generative grammar of Afar
T − 1)−1 .. ··· where nk := #Sk (X). We deduce TSπ1 ×X = (t − 1)− k (−1) kn k = (t − 1)−χ(X) = TS 1 χ(X) . 8 (The torsion of a product). The computations in the above example generalize as follows. Suppose X and Y are compact CW-complexes.
The Euler isomorphism of this acyclic complex induces an isomorphism EulH (A,B,C) : Det(H (A, B, C)) → Det(0). Taking the tensor product of Det(H (A, B, C)) with Det(H∗ (C)) and then applying the Koszul contraction to the pair Det(H∗ (C))−1 , Det(H∗ (C)) we obtain an isomorphism ˆ Det(H∗ (B)) → Det(H∗ (C)). 18. The diagram below is commutative. ˆ Det(B) Det(A) ⊗ EulA ⊗ EulB detf,g w Det(C) EulC u ˆ Det H∗ (A) ⊗ Det H∗ (B) u H (detf,g ) w Det H∗ (C) . 7) To better understand the meaning of the above result suppose we fix bases a, b, c of A, B and respectively C, and bases [a], [b], [c] of H∗ (A), H∗ (B) and respectively H∗ (C).
Is an algebraic contraction. Then Observe that η2 = 0 and ∂η TSπ1 ×X = det ∂ˆ + η : C even (R × X) → C odd (R ⊗ X) . With respect to the bases e0 ⊗ S0 (X) ∪ e1 ⊗ S1 (X) ∪ e0 ⊗ S2 (X) ∪ · · · of C even , and e1 ⊗ S0 (X) ∪ e0 ⊗ S1 (X) ∪ e1 ⊗ S2 (X) ∪ e0 ⊗ S3 (X) ∪ · · · of C odd , the operator ∂ˆ + η has the description n0 columns n1 columns n2 columns · · · n0 rows (t − 1)−1 −∂ 0 ··· n1 rows 0 (t − 1) ∂ ··· n2 rows .. 0 .. 0 .. (t − 1)−1 .. ··· where nk := #Sk (X). We deduce TSπ1 ×X = (t − 1)− k (−1) kn k = (t − 1)−χ(X) = TS 1 χ(X) .
A generative grammar of Afar by Loren Bliese