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A two-dimensional continuum <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073820/p0738201.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073820/p0738202.png" />, in the four-dimensional Euclidean space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073820/p0738203.png" /> such that its [[Homological dimension|homological dimension]] modulo the given <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073820/p0738204.png" /> is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073820/p0738205.png" />. In this sense these continua are  "dimensionally deficient" . L.S. Pontryagin [[#References|[1]]] has constructed surfaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073820/p0738206.png" /> such that their topological product <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073820/p0738207.png" /> is a continuum of dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073820/p0738208.png" />. 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 [[#References|[2]]] a two-dimensional continuum <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073820/p0738209.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073820/p07382010.png" /> has been constructed whose topological square <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073820/p07382011.png" /> is three-dimensional.
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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|homological dimension]] modulo the given $  m = 2 , 3 \dots $
 +
is $  1 $.  
 +
In this sense these continua are  "dimensionally deficient" . L.S. Pontryagin [[#References|[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 [[#References|[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====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  L.S. Pontryagin,  "Sur une hypothèse fundamentale de la théorie de la dimension"  ''C.R. Acad. Sci. Paris'' , '''190'''  (1930)  pp. 1105–1107</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  V.G. Boltyanskii,  "On a theorem concerning addition of dimension"  ''Uspekhi Mat. Nauk'' , '''6''' :  3  (1951)  pp. 99–128  (In Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  P.S. Aleksandrov,  "An introduction to homological dimension theory and general combinatorial topology" , Moscow  (1975)  (In Russian)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  L.S. Pontryagin,  "Sur une hypothèse fundamentale de la théorie de la dimension"  ''C.R. Acad. Sci. Paris'' , '''190'''  (1930)  pp. 1105–1107</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  V.G. Boltyanskii,  "On a theorem concerning addition of dimension"  ''Uspekhi Mat. Nauk'' , '''6''' :  3  (1951)  pp. 99–128  (In Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  P.S. Aleksandrov,  "An introduction to homological dimension theory and general combinatorial topology" , Moscow  (1975)  (In Russian)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
In fact, Pontryagin constructed a sequence of surfaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073820/p07382012.png" />, each of dimension 2, with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073820/p07382013.png" /> <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073820/p07382014.png" />-dimensional, but <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073820/p07382015.png" /> <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073820/p07382016.png" />-dimensional if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073820/p07382017.png" />; and these surfaces exhibit all possibilities in the sense that if a metric continuum <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073820/p07382018.png" /> satisfies <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073820/p07382019.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073820/p07382020.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073820/p07382021.png" /> for all metric continua <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073820/p07382022.png" />. V.G. Boltyan'skii constructed <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073820/p07382023.png" />-dimensional continua <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073820/p07382024.png" /> with the opposite behaviour, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073820/p07382025.png" /> but <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073820/p07382026.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073820/p07382027.png" />; and these surfaces exhibit all possibilities, in the same sense.
+
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|Absolute retract for normal spaces]]; [[Retract of a topological space|Retract of a topological space]]). His examples <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073820/p07382028.png" /> are <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073820/p07382029.png" />-dimensional with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073820/p07382030.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073820/p07382031.png" /> [[#References|[a1]]].
+
Recently A.N. Dranishnikov showed that there even exist dimensionally-deficient absolute neighbourhood retracts (cf. e.g. [[Absolute retract for normal spaces|Absolute retract for normal spaces]]; [[Retract of a topological space|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 $[[#References|[a1]]].
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  A.N. Dranishnikov,  "Homological dimension theory"  ''Russian Math. Surveys'' , '''43''' :  4  (1988)  pp. 11–63  ''Uspekhi Mat. Nauk'' , '''43''' :  4  (1988)  pp. 11–55</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  A.N. Dranishnikov,  "Homological dimension theory"  ''Russian Math. Surveys'' , '''43''' :  4  (1988)  pp. 11–63  ''Uspekhi Mat. Nauk'' , '''43''' :  4  (1988)  pp. 11–55</TD></TR></table>

Latest revision as of 08:07, 6 June 2020


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
How to Cite This Entry:
Pontryagin surface. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Pontryagin_surface&oldid=48245
This article was adapted from an original article by P.S. Aleksandrov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article