ScalingStacks

Lemma 4.3 [014M]

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Lemma 4.3

Let MM be an oriented, compact six-manifold with H2โ€‹(M,๐™)=๐™101H^{2}(M,{\bf Z})={\bf Z}^{101}, and let SS be a subset of 105 elements of H2โ€‹(M,๐™)H^{2}(M,{\bf Z}), named so that

S={Eiโ€‹jl|0โ‰คi<jโ‰ค4,1โ‰คlโ‰ค4}โˆช{Eiโ€‹jโ€‹kl|0โ‰คi<j<kโ‰ค4,1โ‰คlโ‰ค6}โˆช{L0,โ€ฆ,L4}S=\{E^{l}_{ij}|0\leq i<j\leq 4,1\leq l\leq 4\}\cup\{E^{l}_{ijk}|0\leq i<j<k\leq 4,1\leq l\leq 6\}\cup\{L_{0},\ldots,L_{4}\}

Suppose the elements of SS have intersection numbers identical to those given in Proposition 4.2 (2)-(4). Then the set SS generates H2โ€‹(M,๐™)H^{2}(M,{\bf Z}), and there is an isomorphism H2โ€‹(M,๐™)โ‰…H2โ€‹(W,๐™)H^{2}(M,{\bf Z})\cong H^{2}(W,{\bf Z}) preserving the cubic form, taking each divisor in SS to the identically named divisor in H2โ€‹(W,๐™)H^{2}(W,{\bf Z}).

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