Atlas: On linking of cycles in Menger Manifolds by Rolando Jimenez

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1999 Spring Topology Conference
March 18-20, 1999
University of Utah
Salt Lake City, UT, USA

Organizers
Mladen Bestvina, Greg Conner, Misha Kapovich, Bruce Kleiner

On linking of cycles in Menger Manifolds
by
Rolando Jimenez
Instituto de Matematicas, UNAM
Coauthors: Evgeny Shchepin

A cycle z of a space M lying in the complement of a compact set X subset M is called linked with X in M if z is homologically trivial in M and nontrivial in M \X.

In euclidean space the dimension of a compact space is determined by the maximum of dimension of cycles which links with this compact space in some open set. Namely a k-dimensional compact space in a n-dimensional space is locally linked with some cycle of complementary dimension (n-k-1) and it is unlinked with any cycle of less dimension. This statement is known as the P. S. Alexandrov's Theorem on Obstructions in homological dimension theory [1].

The problem we are interested is to find which compacta admits some colinked imbedding into some Menger manifold (See [2] for references). Our main Theorem states.

Theorem If M is a connected n-dimensional Menger manifold then every k-dimensional cycle (k < n) in M is not linked

with any compact space of dimension less than n-k-1

with any polyhedron of dimension less than n-k.

[1] P.S. Alexandrov, Introduction to the homological theory of dimension, Nauka, Moscow, 1975 (Russian)

[2] M. Bestvina, Characterizing k-dimensional universal Menger compacta, Mem. Amer. Math. Soc. 380 Amer. Math. Soc., Providence, RI, 1988

Date received: January 18, 1999