Piecewise Linear Normal Micro-bundles
نویسنده
چکیده
PROOF. We first observe that the pair (M, dM) is (m — k — 1)connected. Indeed, since inclusion induces an isomorphism of fundamental groups, we can use the relative Hurewicz theorem to compute the first nonvanishing relative homotopy group: TTi(M, dM) =TTi(My dM) (where M denotes the universal cover), 7Ti(M, dM) Ç^Hi(M, dM)=HTM~\M), by duality, where c denotes compact cohomology, and H?~\M)^HTM-\K) vanishes for i<m-k. I t follows that for m^2k + l, ƒ is homotopic to a map g: K-+dM. If, now m^2k+2 we can move g into general position (see e.g. [ l l , Chapter 6, Theorem 18]) and so suppose it an imbedding. Take a regular neighbourhood L of g(K) in dM. Then L is a manifold, and the inclusion LQM is a simple homotopy equivalence. If m = 2k + l, g will in general have singularities, transverse selfintersections of fe-simplexes of K. For each such selfintersection Q = g(Pi) =g(P2), we join P i to P 2 by a path a in K such that g (a) is a nullhomotopic loop (since g*: wi(K)—»7Ti(dilf) is onto, this is possible). As fe^3, we can now map a disc D into 9Af, with its interior imbedded, and meeting g(K) only in its boundary, which is attached along g(a). Proceeding thus for each selfintersection Q, we obtain an imbedding of a complex K simply homotopy-equivalent to K; we can then take a regular neighbourhood to obtain L, as above. Note in either case that as regular neighbourhood of a subcomplex K! of codimension ^ 3, L has the property 7TI(3L)=7TIL, for ÔL is a deformation retract of L—K'.
منابع مشابه
Piecewise Linear Transversality
We prove transversality theorems for piecewise linear manifolds, maps and polyhedra. Our main result is that given two closed manifolds contained in a third, then one can be ambient isotoped until it is transversal to the other. This result is then extended to maps and polyhedra. The transversality theory for smooth manifolds was initiated by Thorn in his classical paper [8], and has been exten...
متن کاملThe implications of piecewise linear process of normal accruals
The present study investigates whether the basic assumption in the Jones model, which normal accruals are a linear function of change in sales, is empirically valid. It also discusses and addresses the implications of the assumption violation in the earnings management detection tests. The research employs a sample of 2832 observations of the annual information of firms listed in Tehran Stock E...
متن کاملToric Vector Bundles, Branched Covers of Fans, and the Resolution Property
We associate to each toric vector bundle on a toric variety X(∆) a “branched cover” of the fan ∆ together with a piecewise-linear function on the branched cover. This construction generalizes the usual correspondence between toric Cartier divisors and piecewise-linear functions. We apply this combinatorial geometric technique to study the moduli of toric vector bundles with fixed equivariant Ch...
متن کاملApproximate Solution of Sensitivity Matrix of Required Velocity Using Piecewise Linear Gravity Assumption
In this paper, an approximate solution of sensitivity matrix of required velocity with final velocity constraint is derived using a piecewise linear gravity assumption. The total flight time is also fixed for the problem. Simulation results show the accuracy of the method. Increasing the midway points for linearization, increases the accuracy of the solution, which this, in turn, depends on the...
متن کاملPerturbation analysis of TBR model reduction in application to trajectory-piecewise linear algorithm for MEMS structures
In this paper we use perturbation theory approach to analyze using truncated balanced realization (TBR) linear reduction in a Trajectory-Piecewise linear (TPWL) nonlinear model reduction method. We show that the most important factor affecting perturbation properties of the reduction basis of TBR is a spacing of Hankel singular values. The result is applied to choosing an order of reduction basis.
متن کامل