By Larry Smith

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Additional info for Algebraic topology: Proc. conf. Goettingen 1984

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We can add alternating forms and multiply by scalars so they form a vector space, isomorphic to the space of skew-symmetric n × n matrices. This has dimension n(n − 1)/2, spanned by the basis elements E ab for a < b where Eijab = 0 if ab ab {a, b} = {i, j} and Eab = −Eba = 1. Definition 13 The second exterior power Λ2 V of a finite-dimensional vector space is the dual space of the vector space of alternating bilinear forms on V . Elements of Λ2 V are called 2-vectors. 41 This definition is a convenience – there are other ways of defining Λ2 V , and for most purposes it is only its characteristic properties which one needs rather than what its objects are.

Up ∧ v1 ∧ . . vq and indeed there is. So suppose a ∈ Λp V, b ∈ Λq V , we want to define a ∧ b ∈ Λp+q V . Now for fixed vectors u1 , . . , up ∈ V , M (u1 , u2 , . . , up , v1 , v2 , . . , vq ) is an alternating multilinear function of v1 , . . jq vj1 ∧ . . jq M (u1 , . . , up , vj1 , . .

Then extend to bases {u, u1 } for U1 and {u, u2 } for U2 . The line in P (Λ2 V ) joining L1 and L2 is then P (W ) where W is spanned by u ∧ u1 and u ∧ u2 . 50 Any 2-vector in W is thus of the form λ1 u ∧ u1 + λ2 u ∧ u2 = u ∧ (λ1 u1 + λ2 u2 ) which is decomposable and so represents a point in Q. Conversely, if the lines do not intersect, U1 ∩ U2 = {0} so V = U1 ⊕ U2 . In this case choose bases {u1 , v1 } of U1 and {u2 , v2 } of U2 . Then {u1 , v1 , u2 , v2 } is a basis of V and in particular u1 ∧ v1 ∧ u2 ∧ v2 = 0.