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Transactions of the American Mathematical Society
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On the dimension of the product of two compacta and the dimension of their intersection in general position in Euclidean space

Author(s): A. N. Dranishnikov
Journal: Trans. Amer. Math. Soc. 352 (2000), 5599-5618.
MSC (2000): Primary 55M10, 55N45
Posted: August 8, 2000
MathSciNet review: 1781276
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Abstract | References | Similar articles | Additional information

Abstract:

For every two compact metric spaces $X$ and $Y$, both with dimension at most $n-3$, there are dense $G_{\delta}$-subsets of mappings $f:X \to \mathbb{R}^n$ and $g:Y\to \mathbb{R}^n$ with $dimf(X)\cap g(Y)\leq dim(X\times Y)-n$.

References:

[1]
D. McCullough and L. Rubin Intersection of separators and essential submanifolds of $I^N$, Fund. Math. vol 116 (1983), 131-142. MR 85j:54055

[2]
D. McCullough and L. Rubin Some $m$-dimensional compacta admitting a dense set of imbedding into $\mathbb{R}^{2m}$, Fund. Math. vol 133 (1989), 237-245. MR 91h:54055

[3]
J. Krasinkiewicz and K. Lorentz Disjoint membranes in cubes, Bull. Polish Acad. Sci. Math. 36 (1988), 397-402. MR 92d:54045

[4]
J. Krasinkiewicz Imbeddings into $\mathbb{R}^n$ and dimension of products, Fund. Math. 133 (1989), 247-253. MR 91h:54053

[5]
S. Spiez Imbeddings in $\mathbb{R}^n$ of $m$-dimensional compacta with $\dim(X\times X)<2m$, Fund. Math. 134 (1990), 105-115. MR 91j:54062

[6]
A.N. Dranishnikov and E.V. Scepin Stability of intersections of compact spaces in euclidean space, Russian Mathematical Surveys 44:5 (1989), 194-195. MR 91k:54057

[7]
A.N. Dranishnikov On the mapping intersection problem, Pacif. J. Math. 173 (1996), 403-412. MR 97e:54030
[8]
A.N. Dranishnikov, D. Repovs, E.V. Scepin On intersections of compacta of complementary dimensions in Euclidean space, Topol. Appl. 38 (1991), 237-253. MR 92g:57032
[9]
A.N. Dranishnikov Spanier-Whitehead duality and stability of intersections of compacta, Trudy Mat. Inst. Steklov. 196 (1991), 47-50; English transl., Proc. Steklov Inst. Math. 1992, no. 4 (196), 53-56. MR 92h:55003
[10]
A.N. Dranishnikov On intersection of compacta in Euclidean space, Proc. Amer. Math. Soc. 112 (1991), 267-275. MR 91h:54024

[11]
A.N. Dranishnikov On intersection of compacta in Euclidean space. II, Proc. Amer. Math. Soc. 113 (1991), 1149-1154. MR 92c:54015
[12]
S. Spiez On pairs of compacta with $\dim(X\times Y)<\dim X+\dim Y$, Fund. Math. 135 (1990), 213-222. MR 91j:54064

[13]
J. Segal and S. Spiez On transversaly trivial maps, Q and A in Gen. Topol. 8 (1990), 91-100. MR 91i:54010
[14]
S. Spiez and H. Torunczyk Moving compacta in $\mathbb{R}^n$ apart , Topol. Appl. 41 (1991), 193-204. MR 92m:57030

[15]
A. Dranishnikov, D. Repovs, E. Scepin On intersection of compacta in Euclidean space: the metastable case, Tsukuba J. Math. 17 (1993), 549-564. MR 95a:55002

[16]
A. Dranishnikov and J. West On compacta that intersect unstably in Euclidean space, Topol. Appl. 43 (1992), 181-187. MR 92m:54063

[17]
A.N. Dranishnikov, D. Repovs, E.V.Scepin On approximation and embedding problems for cohomological dimension, Topol. Appl. 55 (1994), 67-86. MR 94m:55001

[18]
W. Olszewski Completion theorem for cohomological dimensions, Proc. Amer. Math. Soc. 123 (1995), 2261-2264. MR 95k:54064

[19]
V.I. Kuzminov Homological dimension theory, Russian Math. Surveys 23, issue 5 (1968), 1-45. MR 39:2158
[20]
A. Dranishnikov Homological dimension theory, Russian Math. Survey 43, issue 4 (1988), 11-63. MR 90e:55003

[21]
J. Dydak Cohomological dimension and metrizable spaces II, Trans. Amer. Math. Soc. 348 (1996). MR 96h:55001

[22]
A.N. Dranishnikov An extension of mappings into CW-complexes, Mat. Sb. 182 (1991), 1300-1310; English transl., Math. USSR Sb. 74 (1993), 47-56. MR 93a:55002.

[23]
A. Dold and R. Thom Quasifaserungen und unendliche symmetrische Produkte, Ann. of Math. 67 (1958), 239-281. MR 20:3542

[24]
M.A. Stanko Embeddings of compacta in Euclidean space, Math. USSR-Sbornik 10 (1970), 234-254. MR 42:6804

[25]
A.N. Dranishnikov Eilenberg-Borsuk theorem for maps in arbitrary complexes, Russian Acad. Sci. Sb. Math. 81 (1995), 467-475. MR 95j:54028

[26]
A.N. Dranishnikov, D. Repovs and E.V. Scepin Dimension of product with continua, Topology Proceedings 18 (1993), 57-73. MR 96b:54054

[27]
A. Dranishnikov and D. Repovs On unstable intersection of 2-dimensional compacta in Euclidean 4-space, Topol. Appl. 54 (1993), 3-11. MR 95a:55001

[28]
A. Dranishnikov and D. Repovs Cohomological dimension with respect to perfect groups, Topol. Appl. 74 (1996), 123-140. MR 98i:55002

[29]
J. Dydak and K. Yokoi Hereditarily aspherical compacta, Proc. Amer. Math. Soc. 124 (1996), 1933-1940. MR 96h:57019

[30]
A. Dranishnikov and J. Dydak Extension dimension and extension types, Proc. Steklov Inst. Math. 212 (1996), 55-88. MR 99h:54049

[31]
Y. Sternfeld A short elementary proof of the Dranishnikov-West theorem on stable intersection of compacta in Euclidean space, Topology Appl. 74 (1996), 177-178. MR 97m:54058

[32]
D. Sullivan Geometric topology, Part I: Localization, Periodicity, and Galois Symmetry, M.I.T. Press (1970). MR 58:13006a

[33]
R.D. Edwards Demension theory I, Lecture Notes in Mathematics 438 (1975), 195-211. MR 52:15477

[34]
R. Engelking Theory of Dimensions Finite and Infinite, Heldermann Verlag (1995). MR 97j:54033

[35]
R. E. Mosher and M.C. Tangora Cohomology operations and applications in homotopy theory, Harper & Row (1968). MR 37:2223

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Additional Information:

A. N. Dranishnikov
Affiliation: Department of Mathematics, Pennsylvania State University, University Park, Pennsylvania 16802
Address at time of publication: Department of Mathematics, University of Florida, Gainesville, Florida 32611
Email: dranish@math.psu.edu, dranish@math.ufl.edu

DOI: 10.1090/S0002-9947-00-02684-2
PII: S 0002-9947(00)02684-2
Received by editor(s): January 30, 1995
Received by editor(s) in revised form: January 27, 1999
Posted: August 8, 2000
Copyright of article: Copyright 2000, American Mathematical Society




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