Approximating topological metrics by Riemannian metrics

Authors:
Steven C. Ferry and Boris L. Okun

Journal:
Proc. Amer. Math. Soc. **123** (1995), 1865-1872

MSC:
Primary 53C23; Secondary 57N60, 57R12

MathSciNet review:
1246524

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We study the relation between (topological) inner metrics and Riemannian metrics on smoothable manifolds. We show that inner metrics on smoothable manifolds can be approximated by Riemannian metrics. More generally, if is a continuous surjection from a smooth manifold to a compact metric space with connected for every , then there is a metric *d* on *X* and a sequence of Riemannian metrics on *M* so that converges to (*X, d*) in Gromov-Hausdorff space. This is used to obtain a (fixed) contractibility function and a sequence of Riemannian manifolds with as contractibility function so that is infinite dimensional. Using results of Dranishnikov and Ferry, this also gives examples of nonhomeomorphic manifolds *M* and *N* and a contractibility function so that for every there are Riemannian metrics and on *M* and *N* so that and have contractibility function and .

**[Be]**Mladen Bestvina,*Characterizing 𝑘-dimensional universal Menger compacta*, Mem. Amer. Math. Soc.**71**(1988), no. 380, vi+110. MR**920964**, 10.1090/memo/0380**[B]**R. H. Bing,*Partitioning continuous curves*, Bull. Amer. Math. Soc.**58**(1952), 536–556. MR**0049550**, 10.1090/S0002-9904-1952-09621-X**[C]**Mark Cassorla,*Approximating compact inner metric spaces by surfaces*, Indiana Univ. Math. J.**41**(1992), no. 2, 505–513. MR**1183356**, 10.1512/iumj.1992.41.41029**[D]**A. N. Dranishnikov,*On resolutions of 𝐿𝐶ⁿ-compacta*, Geometric topology and shape theory (Dubrovnik, 1986) Lecture Notes in Math., vol. 1283, Springer, Berlin, 1987, pp. 48–59. MR**922271**, 10.1007/BFb0081418**[DF]**A. N. Dranishnikov and S. C. Ferry,*Cell-like images of topological manifolds and limits of manifolds in Gromov-Hausdorff space*, preprint.**[DFW]**A. N. Dranishnikov, Steven C. Ferry, and Shmuel Weinberger,*Large Riemannian manifolds which are flexible*, Ann. of Math. (2)**157**(2003), no. 3, 919–938. MR**1983785**, 10.4007/annals.2003.157.919**[DW]**Jerzy Dydak and John J. Walsh,*Infinite-dimensional compacta having cohomological dimension two: an application of the Sullivan conjecture*, Topology**32**(1993), no. 1, 93–104. MR**1204409**, 10.1016/0040-9383(93)90040-3**[F]**Steven C. Ferry,*Constructing 𝑈𝑉^{𝑘}-maps between spheres*, Proc. Amer. Math. Soc.**120**(1994), no. 1, 329–332. MR**1166355**, 10.1090/S0002-9939-1994-1166355-5**[G]**M. Gromov,*Large Riemannian manifolds*, Curvature and topology of Riemannian manifolds (Katata, 1985) Lecture Notes in Math., vol. 1201, Springer, Berlin, 1986, pp. 108–121. MR**859578**, 10.1007/BFb0075649**[KS]**Robion C. Kirby and Laurence C. Siebenmann,*Foundational essays on topological manifolds, smoothings, and triangulations*, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1977. With notes by John Milnor and Michael Atiyah; Annals of Mathematics Studies, No. 88. MR**0645390****[L]**R. C. Lacher,*Cell-like mappings and their generalizations*, Bull. Amer. Math. Soc.**83**(1977), no. 4, 495–552. MR**0645403**, 10.1090/S0002-9904-1977-14321-8**[Mo]**T. Moore,*Gromov-Hausdorff convergence to non-manifolds*, J. Geometric Anal. (to appear).**[P]**Peter Petersen V,*A finiteness theorem for metric spaces*, J. Differential Geom.**31**(1990), no. 2, 387–395. MR**1037407****[S]**Stephen Smale,*A Vietoris mapping theorem for homotopy*, Proc. Amer. Math. Soc.**8**(1957), 604–610. MR**0087106**, 10.1090/S0002-9939-1957-0087106-9**[W]**John J. Walsh,*Isotoping mappings to open mappings*, Trans. Amer. Math. Soc.**250**(1979), 121–145. MR**530046**, 10.1090/S0002-9947-1979-0530046-8

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC:
53C23,
57N60,
57R12

Retrieve articles in all journals with MSC: 53C23, 57N60, 57R12

Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1995-1246524-7

Keywords:
Riemannian manifold,
length space,
cell-like map

Article copyright:
© Copyright 1995
American Mathematical Society