Pontryagin surface

A two-dimensional continuum $ C _ {m} $, $ \mathop{\rm dim} C _ {m} = 2 $, in the four-dimensional Euclidean space $ \mathbf R ^ {4} $ such that its homological dimension modulo the given $ m = 2 , 3 \dots $ is $ 1 $. In this sense these continua are "dimensionally deficient" . L.S. Pontryagin [1] has constructed surfaces $ C _ {2} , C _ {3} $ such that their topological product $ C = C _ {2} \times C _ {3} $ is a continuum of dimension $ 3 $. 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 $ C $ in $ \mathbf R ^ {4} $ has been constructed whose topological square $ C ^ {2} = C \times C $ 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 $ P _ {n} $, each of dimension 2, with $ P _ {n} \times P _ {n} $ $ 4 $- dimensional, but $ P _ {m} \times P _ {n} $ $ 3 $- dimensional if $ m \neq n $; and these surfaces exhibit all possibilities in the sense that if a metric continuum $ X $ satisfies $ \mathop{\rm dim} ( X \times P _ {n} ) = \mathop{\rm dim} X+ 2 $ for all $ n $, then $ \mathop{\rm dim} ( X \times Y) = \mathop{\rm dim} X + \mathop{\rm dim} Y $ for all metric continua $ Y $. V.G. Boltyan'skii constructed $ 2 $- dimensional continua $ B _ {n} $ with the opposite behaviour, $ \mathop{\rm dim} ( B _ {n} \times B _ {n} ) = 3 $ but $ \mathop{\rm dim} ( B _ {m} \times B _ {n} ) = 4 $ for $ m \neq n $; 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 $ D _ {n} $ are $ 4 $- dimensional with $ \mathop{\rm dim} ( D _ {m} \times D _ {n} ) = 7 $ for $ m \neq n $[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