Homotopy and homology groups of the n-dimensional Hawaiian earring

Katsuya Eda; Kazuhiro Kawamura

Fundamenta Mathematicae (2000)

  • Volume: 165, Issue: 1, page 17-28
  • ISSN: 0016-2736

Abstract

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For the n-dimensional Hawaiian earring ℍ n , n ≥ 2, π n ( ℍ n , o ) ≃ ℤ ω and π i ( ℍ n , o ) is trivial for each 1 ≤ i ≤ n - 1. Let CX be the cone over a space X and CX ∨ CY be the one-point union with two points of the base spaces X and Y being identified to a point. Then H n ( X ∨ Y ) ≃ H n ( X ) ⊕ H n ( Y ) ⊕ H n ( C X ∨ C Y ) for n ≥ 1.

How to cite

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Eda, Katsuya, and Kawamura, Kazuhiro. "Homotopy and homology groups of the n-dimensional Hawaiian earring." Fundamenta Mathematicae 165.1 (2000): 17-28. <http://eudml.org/doc/212457>.

@article{Eda2000,
abstract = {For the n-dimensional Hawaiian earring $ℍ_n,$ n ≥ 2, $π _n(ℍ_n,o)≃ ℤ^ω$ and $π_i(ℍ_n, o)$ is trivial for each 1 ≤ i ≤ n - 1. Let CX be the cone over a space X and CX ∨ CY be the one-point union with two points of the base spaces X and Y being identified to a point. Then $H_n(X∨Y) ≃ H_\{n\}(X) ⊕ H_n(Y) ⊕ H_\{n\}(CX∨CY)$ for n ≥ 1.},
author = {Eda, Katsuya, Kawamura, Kazuhiro},
journal = {Fundamenta Mathematicae},
keywords = {homology group; Čech homotopy group; n-dimensional Hawaiian earring},
language = {eng},
number = {1},
pages = {17-28},
title = {Homotopy and homology groups of the n-dimensional Hawaiian earring},
url = {http://eudml.org/doc/212457},
volume = {165},
year = {2000},
}

TY - JOUR
AU - Eda, Katsuya
AU - Kawamura, Kazuhiro
TI - Homotopy and homology groups of the n-dimensional Hawaiian earring
JO - Fundamenta Mathematicae
PY - 2000
VL - 165
IS - 1
SP - 17
EP - 28
AB - For the n-dimensional Hawaiian earring $ℍ_n,$ n ≥ 2, $π _n(ℍ_n,o)≃ ℤ^ω$ and $π_i(ℍ_n, o)$ is trivial for each 1 ≤ i ≤ n - 1. Let CX be the cone over a space X and CX ∨ CY be the one-point union with two points of the base spaces X and Y being identified to a point. Then $H_n(X∨Y) ≃ H_{n}(X) ⊕ H_n(Y) ⊕ H_{n}(CX∨CY)$ for n ≥ 1.
LA - eng
KW - homology group; Čech homotopy group; n-dimensional Hawaiian earring
UR - http://eudml.org/doc/212457
ER -

References

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  1. [1] M. G. Barratt and J. Milnor, An example of anomalous singular theory, Proc. Amer. Math. Soc. 13 (1962), 293-297. Zbl0111.35401
  2. [2] B. de Smit, The fundamental group of the Hawaiian earring is not free, Internat. J. Algebra Comput. 2 (1992), 33-37. Zbl0738.20033
  3. [3] K. Eda, First countability and local simple connectivity of one point unions, Proc. Amer. Math. Soc. 109 (1990), 237-241. Zbl0697.55016
  4. [4] K. Eda, The first integral singular homology groups of one point unions, Quart. J. Math. Oxford 42 (1991), 443-456. Zbl0754.55004
  5. [5] K. Eda, Free σ-products and noncommutatively slender groups, J. Algebra 148 (1992), 243-263. Zbl0779.20012
  6. [6] K. Eda and K. Kawamura, The singular homology of the Hawaiian earring, J. London Math. Soc., to appear. Zbl0958.55004
  7. [7] H. B. Griffiths, The fundamental group of two spaces with a common point, Quart. J. Math. Oxford 5 (1954), 175-190. Zbl0056.16301
  8. [8] H. B. Griffiths, Infinite products of semigroups and local connectivity, Proc. London Math. Soc. 6 (1956), 455-485. Zbl0071.01902
  9. [9] J. W. Morgan and I. Morrison, A Van Kampen theorem for weak joins, Proc. London Math. Soc. 53 (1986), 562-576. Zbl0609.57002
  10. [10] G. W. Whitehead, Elements of Homotopy Theory, Springer, 1978. Zbl0406.55001

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