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a completely general construction of a CW complex which is a classifying space ( B Gamma ) for a finitely presented group (Gamma ), a completely algebraic definition of group cohomology.
(pi _1(S) cong Gamma ), where (Gamma ) is as above, the classifying map (f : S rightarrow X), where X is the Bagnera De Franchis threefold which is a classifying space for (Gamma ), has the property that ( f_* [S] : = Y) satisfies ( Y^3=6).
A Sicilian surface S with (q=p_g=1) is characterized by the following properties: (1) (K_S^2 = 6) (2) ( chi (S) = 1) (3) (pi _1(S) cong Gamma ), where (Gamma ) is as above, (4) the classifying map (f : S rightarrow X), where X is the Bagnera De Franchis threefold which is a classifying space for (Gamma ), has the property that ( f_* [S] : = Y) satisfies ( Y^3=6).
Let BG be a CW-complex which is a classifying space for the group (G = F / R), such that (i) its 1-skeleton (BG^1) has all the 1-cells in bijective correspondence with a set of generators of F, (ii) its 2-skeleton (BG^2) has all the 2-cells in bijective correspondence with a set of generators of R (hence R / [R, R] is isomorphic to the relative homology group (H_2 BG^2, BG^1, {mathbb {Z}}))).
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Justyna Jupowicz-Kozak
CEO of Professional Science Editing for Scientists @ prosciediting.com