Is a space homotopy dominated by a polyhedron P homotopy equivalent to a weak retract of P?
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A space $A$ is homotopy dominated by a space $X$ if there are maps $f:Ato X$ and $g:Xto A$ so that $gcirc fsimeq 1_A$. Also, a subset $A$ of a space $X$ is called a weak retract of $X$ if there exists a map $r:Xto A$ so that $rcirc isimeq 1_A$, where $i:Ato X$ is the inclusion map.
My question is:
Let $A$ be homotopy dominated by a polyhedron $P$. Is there any weak retract $B$ of $P$ so that $A$ and $B$ are homotopy equivalent?
algebraic-topology homotopy-theory
asked Dec 12 '18 at 18:08
M.RamanaM.Ramana
50319
$endgroup$
add a comment |
$begingroup$
A space $A$ is homotopy dominated by a space $X$ if there are maps $f:Ato X$ and $g:Xto A$ so that $gcirc fsimeq 1_A$. Also, a subset $A$ of a space $X$ is called a weak retract of $X$ if there exists a map $r:Xto A$ so that $rcirc isimeq 1_A$, where $i:Ato X$ is the inclusion map.
My question is:
Let $A$ be homotopy dominated by a polyhedron $P$. Is there any weak retract $B$ of $P$ so that $A$ and $B$ are homotopy equivalent?
algebraic-topology homotopy-theory
asked Dec 12 '18 at 18:08
M.RamanaM.Ramana
50319
$endgroup$
$begingroup$
I deleted my answer because your edit made clear what you expect. You should also edit the title of your question: Instead of "correspond to" you should write "homotopy equivalent to".
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– Paul Frost
Dec 13 '18 at 11:44
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@PaulFrost Thank you for your answer. Yes, sure. You are right.
$endgroup$
– M.Ramana
Dec 14 '18 at 17:19
add a comment |
$begingroup$
A space $A$ is homotopy dominated by a space $X$ if there are maps $f:Ato X$ and $g:Xto A$ so that $gcirc fsimeq 1_A$. Also, a subset $A$ of a space $X$ is called a weak retract of $X$ if there exists a map $r:Xto A$ so that $rcirc isimeq 1_A$, where $i:Ato X$ is the inclusion map.
My question is:
Let $A$ be homotopy dominated by a polyhedron $P$. Is there any weak retract $B$ of $P$ so that $A$ and $B$ are homotopy equivalent?
algebraic-topology homotopy-theory
asked Dec 12 '18 at 18:08
M.RamanaM.Ramana
50319
$endgroup$
A space $A$ is homotopy dominated by a space $X$ if there are maps $f:Ato X$ and $g:Xto A$ so that $gcirc fsimeq 1_A$. Also, a subset $A$ of a space $X$ is called a weak retract of $X$ if there exists a map $r:Xto A$ so that $rcirc isimeq 1_A$, where $i:Ato X$ is the inclusion map.
My question is:
Let $A$ be homotopy dominated by a polyhedron $P$. Is there any weak retract $B$ of $P$ so that $A$ and $B$ are homotopy equivalent?
algebraic-topology homotopy-theory
algebraic-topology homotopy-theory
asked Dec 12 '18 at 18:08
M.RamanaM.Ramana
50319
asked Dec 12 '18 at 18:08
M.RamanaM.Ramana
50319
edited Dec 14 '18 at 17:19
M.Ramana
asked Dec 12 '18 at 18:08
M.RamanaM.Ramana
50319
asked Dec 12 '18 at 18:08
M.RamanaM.Ramana
50319
asked Dec 12 '18 at 18:08
M.RamanaM.Ramana
50319
50319
$begingroup$
I deleted my answer because your edit made clear what you expect. You should also edit the title of your question: Instead of "correspond to" you should write "homotopy equivalent to".
$endgroup$
– Paul Frost
Dec 13 '18 at 11:44
$begingroup$
@PaulFrost Thank you for your answer. Yes, sure. You are right.
$endgroup$
– M.Ramana
Dec 14 '18 at 17:19
add a comment |
$begingroup$
I deleted my answer because your edit made clear what you expect. You should also edit the title of your question: Instead of "correspond to" you should write "homotopy equivalent to".
$endgroup$
– Paul Frost
Dec 13 '18 at 11:44
$begingroup$
@PaulFrost Thank you for your answer. Yes, sure. You are right.
$endgroup$
– M.Ramana
Dec 14 '18 at 17:19
$begingroup$
I deleted my answer because your edit made clear what you expect. You should also edit the title of your question: Instead of "correspond to" you should write "homotopy equivalent to".
$endgroup$
– Paul Frost
Dec 13 '18 at 11:44
$begingroup$
I deleted my answer because your edit made clear what you expect. You should also edit the title of your question: Instead of "correspond to" you should write "homotopy equivalent to".
$endgroup$
– Paul Frost
Dec 13 '18 at 11:44
$begingroup$
@PaulFrost Thank you for your answer. Yes, sure. You are right.
$endgroup$
– M.Ramana
Dec 14 '18 at 17:19
$begingroup$
@PaulFrost Thank you for your answer. Yes, sure. You are right.
$endgroup$
– M.Ramana
Dec 14 '18 at 17:19
add a comment |
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$begingroup$
I deleted my answer because your edit made clear what you expect. You should also edit the title of your question: Instead of "correspond to" you should write "homotopy equivalent to".
$endgroup$
– Paul Frost
Dec 13 '18 at 11:44
$begingroup$
@PaulFrost Thank you for your answer. Yes, sure. You are right.
$endgroup$
– M.Ramana
Dec 14 '18 at 17:19