ファイバー空間の切断定理 （英文）
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(2) Vol. 10, No. 2 Journal of Hokkaido Gakugei University Dec. 1959. Cross-section Theorems for Fiber Spaces. Nobuo KIMURA The Study of Mathematics, Hakodate Branch, Hokka'ido Gakugei University. ^m^: 7 ^ -f ^-SF^O w^-^s. Recently, Edward Fadell (1) has defined a fiber space which is invariant under fiberhomotopy equivalences, and has proved a cross-section theorem in the case of contractible fibers. In the present paper we shall show that the contractibility condition for fibers can be weakened considerably, and that Fadell's theorem holds for fiber spaces whose base spaces are finite complexes or weakly locally contractible spaces. 1. Definitions and notations which are used in the following are almost the same as in. (1); here we shall recall only the definition of fiber spaces. Definition. Let p: X-^B be a map (we consider only continuous maps) from a space X onto a space 5, and let F be a space ; then the triple (X, B, p^ is called a fiber space if there exist an open covering {Ua} of B and maps ^ : P~l(Ua)^-UaXF : <fta such that </'a0a~l and 0a^a~l are projection-preserving homotopies, i.e. there exist (a) a homotopy H connecting 4)a^a and 1, such that PH(x, i) = x 0 ^i=l, where P is a projection UctXF—fUa, and (b) a homotopy G connecting ^a</'a and 1, such that PG{x, t) -= x 0 ^i^l. 2. We consider a fiber space {X, B, p), and assume, throughout this paper, that any closed subsets of B has the homotopy extension property relative to X. Let / : A—fX be a given cross-section over a closed subset A of 2?. Then, according to E. Fadell, we obtain the following theorem. Theorem. Suppose that the fiber F is contractible and B is compact Hausdorff. Then there exists a cross-section g : B—fX such that g\A^-f preserving projections, 3. Let us consider the following property (P). Property (P). There exists an open covering {Va} of B such that V'adUa, and such that for any closed subset of Va and for any map / : Da-^F there exists a map // : V—fF such that f'\Da~f in F. Theorem. If B is compact Hausdorff and has Property (P), then we have the cross-section theorem in the sense of Fadell's. Proof. By the assumption on B we can find a covering {Va} of B satisfying the con-. 1—.
(3) Nobuo Kimura ditions stated in Property (P). For every Ucc there exists maps <pa, (/)cc such that ^ : p~l (Ua)^lUaXF : (fia. Set AHV^==Da, then DaC.Va, and ^a/ is a map Da-^F where ^ is the projection Ua x F->F.. Since Property (P) holds, there exists a map c : V^F such that K\Da~P4^f in F. For b^Va, define g, : Va^X by. g,(b}=^.{b, K[b)\ 0a^a~l being fiber-homotopy equivalence, we can find a map F : p~l(Ua)x l—>-p~l{Uct) (preserving projections). such that F(x, 0) =x, F(x, 1)= <t>^»{x). Denote by A : D a x I—>-F a map such that. A{b, 0)=^/(&), A{b, l)=K(b}. Define G : D^xI-^X by. G(b, t) = F(f{b\ 2t) O^t^l/2 = <p^(b, A(b, 2t-l)) 1/2 ^ t ^ 1. Then we obtain. <f,a{b, A(b, 0)) = ^a(&, P^ccAb)) =<p^ccf{b) = F(/(6), 1). Hence G{b, i) is continuous over D a xl.. Evidently we have. G(b, 0)=F(/(Z>), 0) =fW, G(b, 1) = 0a(6, yl(6, 1)) = ^(&, /c(6)) = g,(b), and it follows that G(b, l)=g^\Da. According to the homotopy extension property of B relative to X, G can be extended to a map G : VccXl-^X. such that G(b, 1) = <7,(&). Let pG-=H. Setting Hf{b, t}=b for any b€A, extend H to a H/ of (AU^XZ. Now, define g* : A[^Va—>-X as a map. ^=/ on A gtk = G^VaXO on V'a.. We now apply the generalized covering homotopy theorem of (1) using g* and H'. Then there exists a homotopy. G* : {A^V^I-^X.
(4) Cross-secfcion Theorems for Fiber Spaces. such that pG* is strongly homotopic to H/ and G^KAUV'^xO =g*. Let g denote the map G* |(A)JV")xl; then g becomes a cross-section over Aj Ua and 0f|A~/ is a projection-preserving homotopy. Continuing the above processes for a finite number of 17aS which covers B, we obtain a required cross-section.. Remark 1. The case in (1) where F is contractible is a special case of our theorem. Remark 2. When B is a Co-space, we have also an analogous theorem, and its proof. is quite parallel to that of (1). Remark 3. When F is a solid space (2), strong cross-secdon theorem is valid for fiber. bundles is (2) p. 55. Corollary 1. If, in place of assumptions of Theorem, we assume that B is a finite complex, then the cross-section theorem holds too. Proof. We may assume that any cell on is contained in some V^. Let An^n=C. Let Xo be a point of C. Since <?" is contractible to Xy, there exists its contraction r : <jnxl—>-an,. where we have r(x, 0) = a, r(a3, 1) = xo.. For a map K : Vu—fF in the proof of Theorem we now take K(X~) -= P<Paf{X,}.. Then /.- I C~P^af, hence Property (P) holds and Corollary 1 follows immediately from the above theorem. Definition. A space B is called weakly locally contractible, if for every bycB there exists a neighborhood of bo which is contractible to by in B. Corollary 2. The cross-section theorem holds for a fiber space whose base space is compact, arcwise connected and weakly locally contractible. Proof. We may assume that Ua. is contractible to its point. Since B is arcwise connected, Va is contractible in B to any point of its own. Let r : VaXl-^Va be its contraction, hence r(x, 0) .= x, r(x, 1) -= Xa,. (a;o may be any point in Van A) then, proceeding as in the same manner as Corollary 1 we see that Corollary 2 holds. Remark. If we define a weakly locally contractible space as a space such that U is contractible in the whole space to its arbitrary point, then the assumption of arcwise connectedness in Corollary 2 is unnecessary. And the weak local contractibility in this sense is weaker than that we have defined before. References 1) E. Fadell: On fiber spaces, Trans. Amer. Math. Soc. vol. 90 (1959), pp. 1-14.. 2) N. Steenrod: The topology of fibre bundles, Princeton University Press.. 3—.
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