Mixing
Connected Hausdorff space all whose $n$-th homology groups, for $nge 1$, are trivial
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Let $X$ be a connected Hausdorff space such that $H_n(X,mathbb Z)=0, forall n ge 1$. Then does that necessarily imply that $X$ is path connected i.e. $H_0(X,mathbb Z)=mathbb Z$ ?
algebraic-topology homology-cohomology homotopy-theory
asked Jan 10 at 8:17
user521337user521337
1,1451416
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add a comment |
$begingroup$
Let $X$ be a connected Hausdorff space such that $H_n(X,mathbb Z)=0, forall n ge 1$. Then does that necessarily imply that $X$ is path connected i.e. $H_0(X,mathbb Z)=mathbb Z$ ?
algebraic-topology homology-cohomology homotopy-theory
asked Jan 10 at 8:17
user521337user521337
1,1451416
$endgroup$
add a comment |
$begingroup$
Let $X$ be a connected Hausdorff space such that $H_n(X,mathbb Z)=0, forall n ge 1$. Then does that necessarily imply that $X$ is path connected i.e. $H_0(X,mathbb Z)=mathbb Z$ ?
algebraic-topology homology-cohomology homotopy-theory
asked Jan 10 at 8:17
user521337user521337
1,1451416
$endgroup$
Let $X$ be a connected Hausdorff space such that $H_n(X,mathbb Z)=0, forall n ge 1$. Then does that necessarily imply that $X$ is path connected i.e. $H_0(X,mathbb Z)=mathbb Z$ ?
algebraic-topology homology-cohomology homotopy-theory
algebraic-topology homology-cohomology homotopy-theory
asked Jan 10 at 8:17
user521337user521337
1,1451416
asked Jan 10 at 8:17
user521337user521337
1,1451416
asked Jan 10 at 8:17
user521337user521337
1,1451416
asked Jan 10 at 8:17
user521337user521337
1,1451416
asked Jan 10 at 8:17
user521337user521337
1,1451416
1,1451416
add a comment |
add a comment |
1 Answer
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No : take $X$ to be the topologist's sine curve. Then $X$ us connected but not path connected.
Moreover by compactness, any $n$-simplex of $X$ either factors through the usual $mathbb{R}^*_+ to X$ or the point ${*}to X$, and $H_n (X) = 0, ngeq 1$.
answered Jan 10 at 8:32
MaxMax
14.2k11142
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1 Answer
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1 Answer
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active
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active
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active
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votes
$begingroup$
No : take $X$ to be the topologist's sine curve. Then $X$ us connected but not path connected.
Moreover by compactness, any $n$-simplex of $X$ either factors through the usual $mathbb{R}^*_+ to X$ or the point ${*}to X$, and $H_n (X) = 0, ngeq 1$.
answered Jan 10 at 8:32
MaxMax
14.2k11142
$endgroup$
add a comment |
$begingroup$
No : take $X$ to be the topologist's sine curve. Then $X$ us connected but not path connected.
Moreover by compactness, any $n$-simplex of $X$ either factors through the usual $mathbb{R}^*_+ to X$ or the point ${*}to X$, and $H_n (X) = 0, ngeq 1$.
answered Jan 10 at 8:32
MaxMax
14.2k11142
$endgroup$
add a comment |
$begingroup$
No : take $X$ to be the topologist's sine curve. Then $X$ us connected but not path connected.
Moreover by compactness, any $n$-simplex of $X$ either factors through the usual $mathbb{R}^*_+ to X$ or the point ${*}to X$, and $H_n (X) = 0, ngeq 1$.
answered Jan 10 at 8:32
MaxMax
14.2k11142
$endgroup$
No : take $X$ to be the topologist's sine curve. Then $X$ us connected but not path connected.
Moreover by compactness, any $n$-simplex of $X$ either factors through the usual $mathbb{R}^*_+ to X$ or the point ${*}to X$, and $H_n (X) = 0, ngeq 1$.
answered Jan 10 at 8:32
MaxMax
14.2k11142
answered Jan 10 at 8:32
MaxMax
14.2k11142
answered Jan 10 at 8:32
MaxMax
14.2k11142
answered Jan 10 at 8:32
MaxMax
14.2k11142
14.2k11142
add a comment |
add a comment |
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