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Session 37. Wild algebraic and geometric topology
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Singular homology groups of one dimensional Peano continua |
Katsuya Eda, Waseda University, Japan
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Peano continua are locally connected, connected, compact metric spaces.
As we have proved previously, the fundamental groups
of one-dimensional Peano continua determine their homotopy types
[2], and in particular the fundamental groups of one-dimensional
Peano continua which are not semi-locally simply connected everywhere determine
their homeomorphism types [1]. Consequently,
the fundamental groups of one-dimensional Peano continua are abundant.
In contrast to the case of the fundamental groups, we have
Theorem:
Let \(X\) be a one-dimensional Peano continuum. Then the singular homology
group \(H_1(X)\) is isomorphic to a free abelian group of finite rank or
the singular homology group of the Hawaiian earring.
Therefore the classification is the same as the Čech homology groups
and the shape groups. But its proof is very group theoretic and far from a geomeric
one.
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References-
K. Eda, The fundamental groups of one-dimensional spaces and spatial
homomorphisms , Topology Appl. 123, 2002, 479-505.
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K. Eda, Homotopy types of one-dimensional Peano continua , Fund. Math.
209, 2010, 27-45.
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