Pontryagin surface
A two-dimensional continuum ,
, in the four-dimensional Euclidean space
such that its homological dimension modulo the given
is
. In this sense these continua are "dimensionally deficient" . L.S. Pontryagin [1] has constructed surfaces
such that their topological product
is a continuum of dimension
. Thus, the conjecture stating that under topological multiplication of two (metric) compacta their dimensions are added, was disproved. He proved this conjecture for homological dimensions modulo a prime number and, in general, over any group of coefficients which is a field. In [2] a two-dimensional continuum
in
has been constructed whose topological square
is three-dimensional.
References
[1] | L.S. Pontryagin, "Sur une hypothèse fundamentale de la théorie de la dimension" C.R. Acad. Sci. Paris , 190 (1930) pp. 1105–1107 |
[2] | V.G. Boltyanskii, "On a theorem concerning addition of dimension" Uspekhi Mat. Nauk , 6 : 3 (1951) pp. 99–128 (In Russian) |
[3] | P.S. Aleksandrov, "An introduction to homological dimension theory and general combinatorial topology" , Moscow (1975) (In Russian) |
Comments
In fact, Pontryagin constructed a sequence of surfaces , each of dimension 2, with
-dimensional, but
-dimensional if
; and these surfaces exhibit all possibilities in the sense that if a metric continuum
satisfies
for all
, then
for all metric continua
. V.G. Boltyan'skii constructed
-dimensional continua
with the opposite behaviour,
but
for
; and these surfaces exhibit all possibilities, in the same sense.
Recently A.N. Dranishnikov showed that there even exist dimensionally-deficient absolute neighbourhood retracts (cf. e.g. Absolute retract for normal spaces; Retract of a topological space). His examples are
-dimensional with
for
[a1].
References
[a1] | A.N. Dranishnikov, "Homological dimension theory" Russian Math. Surveys , 43 : 4 (1988) pp. 11–63 Uspekhi Mat. Nauk , 43 : 4 (1988) pp. 11–55 |
Pontryagin surface. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Pontryagin_surface&oldid=48245