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16 OPERATIONS ON VECTOR BUNDLES 117

such that w_y o ($' © $") = _w(, o _s' + wjj o _s". We write u = u' + w". If G', G"
are vector bundles over B and if v' : G' -»E' and v" : G" -»E" are B-mor-
phisms, we denote by v' © v" the morphism (/ o v') + (/' o z/') of G' © G" into
E' © E". For each b e B we have (u' + u\ = u'_b + < , (t/ © i/')* = ^ ® *# •

Finally we remark that the fibration (E' © E", B, <r) is isomorphic to the
fiber product E' X_B E" defined in (16.12.10).

(16.16.2) The definitions of Whitney sum and tensor product generalize
immediately to the case of any finite number of vector bundles over B. In
particular, for each integer m > 0, we define a multiple mE (resp. a tensor
power E®^m) of a vector bundle E as the sum (resp. tensor product) of m copies
of E. We use an analogous notation for B-morphisms.

Likewise, for each integer m > 0, the exterior power /\E is defined: this

space has for its underlying set the disjoint union of the sets /\ E_b as b runs

m m

through B, and its projection A : /\ E -» B sends each element of /\ E_b to the
point b e B. For each sequence ($₇-)i^^_m of m sections of E over an open set
U in B, we denote by s_l A s₂ A • • • A $_m the mapping

b^s^b) A s₂(b) A • • - A s_m(A);

the vector bundle structure on /\ E is defined by the condition that, for each
sequence ($,•)! ^^_m of m sections of class C°° of E over an open subset U of B,

the mapping $_x A $₂ A • • • A $_m is a C°°-section of /\ E. If n is the projection
E -» B, and if we are given an open covering (U_a) of B together with diffeo-
morphisms <p_a : U_a x F_a-^7t"¹(\J_a) satisfying (VB), then the fiber bundle

/\E may be constructed as follows: we take the transition homomorphisms
(b, t)\-+(b₉fp_a(b, t)) corresponding to the <p_a, and we form the mappings

(b, t)*-+(b, g^(b₉ 1)), where for each b e U_a n U, , g,Jtb, 0 = A fn*(^b> ')•
If (o^i ^ i^_n is a frame of E over an open set U c B, then the I I sections

of /\ E such that i_l < 1₂ < - - • < i_m form a frame of /\ E over U.

If u : E -* F is a B-morphism of vector bundles, there exists a unique B-

mm m

morphism /\ u : /\ E -* /\ F such that, if Sj , . . . , $_m are sections of E over U,
we have

m
(/\W) o (_$1 A S₂ A " • ' A $J = (W o _$1) A (W o $₂) A • • ' A(W o S_m).



	16 OPERATIONS ON VECTOR BUNDLES 117 such that w_y o ($' © $") = _w(, o _s' + wjj o _s". We write u = u' + w". If G', G" are vector bundles over B and if v' : G' -»E' and v" : G" -»E" are B-mor- phisms, we denote by v' © v" the morphism (/ o v') + (/' o z/') of G' © G" into E' © E". For each b e B we have (u' + u\ = u'_b + < , (t/ © i/')* = ^ ® # • Finally we remark that the fibration* (E' © E", B, <r) is isomorphic to the fiber product E' X_B E" defined in (16.12.10). (16.16.2) The definitions of Whitney sum and tensor product generalize immediately to the case of any finite number of vector bundles over B. In particular, for each integer m > 0, we define a multiple mE (resp. a tensor power E®^m) of a vector bundle E as the sum (resp. tensor product) of m copies of E. We use an analogous notation for B-morphisms. m Likewise, for each integer m > 0, the exterior power /\E is defined: this m space has for its underlying set the disjoint union of the sets /\ E_b as b runs m m through B, and its projection A : /\ E -» B sends each element of /\ E_b to the point b e B. For each sequence ($₇-)i^^_m of m sections of E over an open set U in B, we denote by s_l A s₂ A • • • A $_m the mapping b^s^b) A s₂(b) A • • - A s_m(A); m the vector bundle structure on /\ E is defined by the condition that, for each sequence ($,•)! ^^_m of m sections of class C°° of E over an open subset U of B, m the mapping $_x A $₂ A • • • A $_m is a C°°-section of /\ E. If n is the projection E -» B, and if we are given an open covering (U_a) of B together with diffeo- morphisms <p_a : U_a x F_a-^7t"¹(\J_a) satisfying (VB), then the fiber bundle m /\E may be constructed as follows: we take the transition homomorphisms (b, t)\-+(b₉fp_a(b, t)) corresponding to the <p_a, and we form the mappings (b, t)-+(b, g^(b₉* 1)), where for each b e U_a n U, , g,Jtb, 0 = A fn(^b>* ')• If (o^i ^ i^_n is a frame of E over an open set U c B, then the I I sections

	of /\ E such that i_l < 1₂ < - - • < i_m form a frame of /\ E over U. If u : E -* F is a B-morphism of vector bundles, there exists a unique B- mm m morphism /\ u : /\ E -* /\ F such that, if Sj , . . . , $_m are sections of E over U, we have m (/\W) o (_$1 A S₂ A " • ' A $J = (W o _$1) A (W o $₂) A • • ' A(W o S_m).