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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

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Dimension and Connectivity Coding
by
Evgeny Shchepin
Inst. de Mat. UNAM, Cuernavaca

The dimension code of a compact space as well as the connectivity code of a CW-complex can be represented as sequences of triplets \alpha0, \alpha2, \alpha3, \alpha5 ... indexed by prime numbers. The set of triplets is the following:
0, 1, 1+, 2-, 2, 2+, 3-, 3, 3+, ... n-, n, n+, ...
Ignoring of sign produces from a triplet is natural number calling by its base. Triplets without sign are called regular. Regular triplets one identifies with natural numbers. Triplets with the sign "-" are called minor, and with sign "+" are called major. The are two operation of addition on triplets compatible with the order: the minor sum "[+]" and the major sum "(+). For both operations the base of the sum is equal to the sum of the bases of summands. The type of the sum (minor, major or regular) is defined by the following rules:

  1. If the summands have the same type then its minor/major sum has the same type

  2. If one of the summands is regular, then the type of minor/major sum is the same as the type of other summand

  3. If one summand is of minor type, and the other of major one then the type of sum is defined as minor for the minor operation and as major for the major one.

If \alpha0, \alpha2, \alpha3, \alpha5 ... and \beta0, \beta2, \beta3, \beta5 ... are dimensional codes of compacta X and Y then the dimension code of its product is \alpha0[+]\beta0, \alpha2[+]\beta2, \alpha3[+]\beta3, .... If this codes are connectivity codes of complexes then the connectivity code of its smash-product is defined as \alpha0(+)\beta0, \alpha2(+)\beta2, \alpha3(+)\beta3, ....

The dimension code \alpha0, \alpha2, \alpha3, \alpha5 ... of a compactum contains all necessary information to define its cohomological dimension with respect to any coefficient. For example the dimension modulo p is base of \alphap if p is a positive prime, and the rational dimension coincides with \alpha0. The same is true for the connectivity code. Where by connectivity of a complex K with respect to a coefficient group G one means the first dimension n for which the reduced homology group Hn (K, G) is nontrivial. The dimension/connectivity code \alpha0, \alpha2, \alpha3, \alpha5 ... of any compact/complex satisfies the following conditions:

  1. \alpha0 is regular,

  2. if \alphap is regular, then \alphap = \alpha0,

  3. if \alpha0 = 0, then \alphap = 0 for all p.
And for every sequence of triplet codes \alpha0, \alpha2, \alpha3, \alpha5 ..., which satisfies the above conditions there exist a compact and a complex which dimensional and connectivity codes are expressed by the sequence.

The described theory represents a new form for well known Bockstein Coefficient theory and its dual developed by A. N. Dranishnikov and J. Dydak in . The detailed exposition of the whole theory is contained in .

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A. Dranishnikov and J. Dydak, Ëxtension dimension and extension types", Proc. Steklov Math. Inst., 212, (1996), 61-94.

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E. V. Shchepin, Ärithmetic of the Dimension Theory", Russ. Math. Surv. 221, (1998) iss. 5.

Date received: January 20, 1999

Copyright © 1999 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # caca-14.