Showing papers on "Frame bundle published in 1972"

On the Behavior of Extensions of Vector Bundles Under the Frobenius Map

Abstract: Let k be an algebraically closed field of characteristic p > 0, and let X be a curve defined over k. The aim of this paper is to study the behavior of the Frobenius map F*: H1(X, E) → H1(X, F*E) for a vector bundle E.

The signature of fiber bundles

Abstract: Let Y-*Z-* X be a locally trivial fiber bundle in the category of oriented topological manifolds. It is shown that if the identity component of the structure group G has finite index, then (signature of Z) = (signature of X) * (signature of Y). Let F-*EAB be a locally trivial fiber bundle such that (1) E, F, B are closed, oriented topological manifolds. (2) E, F, B are coherently oriented, that is, the orientation of F and B determine that of E. In this situation, does it follow that a(E)=a(B).a(F), where a( ) denotes the signature homomorphism? That additional conditions are necessary is shown both by Kodaira [4] and Atiyah [1] when they produce a locally trivial fibering of a complex surface by a complex surface such that the total space has a nonzero signature. In fact, in the smooth case, Atiyah produces a formula computing r(E) and showing the dependency on the fundamental group of B. The approach of this paper is to look at the structure group G of the bundle and determine conditions on G in order to obtain an affirmative answer to the above question. If G is any topological group, let r=G/Go, where Go is the connected component of the identity. The main result is the THEOREM. Let G be a locally compact, finite dimensional topological group such that I Fl is finite. If FEA. B is any oriented locally trivial topologicalfiber bundle with structure group G, then aE= oiB aF. REMARK 1. The theorem obviously remains valid if the structure group of the bundle is not, a fortiori, G, but can be reduced to G. REMARK 2. The hypothesis that G be locally compact, finite dimensional only exists to insure that G-r possesses a local cross section. Any other hypothesis on G insuring this is equally valid. See [3], for instance. PROPOSITION 1. If F = (e), then qE= oB * oF. Received by the editors March 4, 1970 and, in revised form, July 20, 1971. AMS 1969 subject class flcations. Primary 5730, 5560.

Tensor fields and connections on a cross-section in the tangent bundle of order r

Abstract: Let M be an ^-dimensional differentiable manifold and Tr(M) the tangent bundle of order r over M, ri^l being an integer [1], [3], [4]. The prolongations of tensor fields and connections given in the differentiable manifold M to its tangent bundle of order r have been studied in [1J, [2], [3] [4], [7], [8] and [9]. If V is a vector field given in M, V determines a cross-section in Tr(M). For the cases r=l and r=2, Yano [7] and Tani [5] have studied, on the cross-section determined by a vector field F, the behavior of the prolongations of tensor fields and connections in M to T(M) (i.e., Ά(M)) and Γa(M), respectively. The purpose of this paper is to study, on the cross-section determined by a vector field V, the behavior of the prolongations of these geometric objects in M to Tr(M) ( r^ l ) . In §1 we summarize the results and properties we need concerning the prolongations of tensor fields and connections in M to Tr(M). Proofs of the statements in §1 can be found in [1], [2], [3], [4] and [8]. In § 2 we study the cross-section determined in Tr(M) by a given vector field V in M In § 3 we study the behavior of prolongations of tensor fields on the cross-section. In §4 we study the prolongations of connections given in M to Tr{M) along the cross-section and some of their properties. We assume in the squel that the manifolds, functions, tensor fields and connections under consideration are all of differentiability of class C°°. Several kinds of indices are used as follows: The indices λ, μ, v, • ••, s, t, u, ••• run through the range 0,1, 2, ••• r; the indices h, i, j , k, m, ••• run through the range 1, 2, ••• n. Double indices like {v)h are used, where O^i^r, l^h^n. The indices Λ, B, C, ••• run through the range (1)1, (1)2, •••, (l)n, (2)1, •••, (2)n, •••, (r)l, •••, (r)n. For a given function / on M, the notation / ( 0 ) is sometimes substituted by f° for simplicity. Summation notation Σ J β l with respect to h> i, j , k, m, ••• (=1, 2, ••• n) is omitted while summation notation with respect to λ, μ, v, •••, s, t, u •••, from 0 to r, will be kept. For example,

A note on the Whitehead torsion of a bundle modulo a subbundle

Abstract: The purpose of this note is to show how the sum theorem for Whitehead torsion due to K. W. Kwun and R. H. Szczarba, and generalized by L. C. Siebenmann, may be applied to compute the Whitehead torsion of the total space pair of a bundle pair in terms of the Whitehead torsion of the fiber pair and the Euler characteristic of the base. Let p:E-+B be a PL fiber bundle with fiber F and suppose p':E'->-B is a PL subbundle with fiber F'. If the inclusion F'c F is a homotopy equivalence, a theorem of Dold [1, Theorem 6.3] implies that the inclusion £<=£' ¡s also a homotopy equivalence. It is the object of this note to answer the natural question1 "How are the Whitehead torsions tíF, F')e Wh tt^F) and t(£, £')eWh «i(£) related?" Specifically we prove the Theorem. Let B and F be connected. Then r(£, £') = XiB)MiF, F') where %iB) is the Euler characteristic of B and j*: Wh 771(F)^-Wh 77,(£) is induced by the inclusion y.F^-E. To set the context precisely, we recall that a PL bundle pair with fiber (F, F') is a PL map p: (£, £')->-(£, B), denoted simply by B in the sequel, such that for some triangulation K of B and each simplex aeK, there is a PL homeomorphism ha:iaxF,axF')-^-ip~1ia),p~1ia)C\E') such that pha=P\ where p^.iaxF, axF')->a is projection on the first factor. A PL bundle p':E'-*B with fiber F' is a subbundle ,of the PL bundle p: E-*B with fiber F if £'<= E,p'=p | £', and />:(£, E')-*B is a PL bundle pair with fiber (£, £'). In this event for each xeB we denote the pair ip-1ix),p'-1ix))=ip~1ix),p~1ix)r\E') by iFx,F'x) and the inclusion (F„F;)c (£,£') by;,. Throughout the remainder of this note we work in the category of compact PL spaces and PL maps. Received by the editors June 1, 1971. AMS 1970 subject classifications. Primary 57C10; Secondary 57C50.

The Tangent Bundle of a Topological Manifold

Abstract: (1972). The Tangent Bundle of a Topological Manifold. The American Mathematical Monthly: Vol. 79, No. 10, pp. 1090-1096.

Skew connections in vector bundles and their prolongations

Abstract: The paper is closely related to [1] and [2]. A skew connection in a vector bundle E as defined here is a pseudo-connection (in the sense of [1]) which can be changed into a connection by transforming separately the bundle E itself and the bundle of its differentials, i.e. one-forms on the base with values in E . The properties of skew connections are thus expected to be only “algebraically” more complicated than those of connections; especially one can follow the pattern of [1], and prolong them to obtain higher order semi-holonomic and non-holonomic pseudo-connections. It is shown in this paper that under some circumstances the main theorem of [1] or [2] applies also to skew connections.

Homology tangent bundles and universal bundles

Abstract: We find results about the evaluation map from the group of homeomorphisms of a closed manifold M and also about fibre bundles where M is the fibre. These facts follow from the observation that the homology tangent bundle is induced from a universal bundle pair.