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Show the projective space $RP^2$ minus a point is homotopy equivalent to the unit circle $S^1$.

I can image how to do by the graph,as think of $RP^2$ as the unit disk with opposite boundary points identified, and remove one point $x$ from this disk, then remaining points can be gradually pushed out to the boundary, and the boundary is a strong deformation retract of $RP^2-\{x\}$, also the the boundary is homeomorphic to $S^1$.

However, I would like to do this by formally defining a homotopy that show $RP^2-\{x\}$ are homotopy equivalent to $S^1$. How would the function looks like?

Any help is appreciated, thanks a lot!

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    $\begingroup$ You mean homotopy equivalent. The spaces are not homeomorphic: for instance, a two-dimensional disk embeds in the one but not the other. $\endgroup$ Apr 2 '15 at 0:31
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    $\begingroup$ @KevinCarlson, yes, I mean homotopy equivalent. thank you. $\endgroup$
    – user138017
    Apr 2 '15 at 0:34
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The easiest way to define a function precisely is via coordinates, and $RP^2$ has a very convenient system of coordinates, called homogeneous. Identifying a point $\ell$ in $RP^2$ with a line $\{v:v\in \ell\}$ in $\mathbb R^3$, take some $v=(v_1,v_2,v_3)\in \ell$. Then $[v_1:v_2:v_3]$ are coordinates of $\ell$ as a point in the projective plane, unique up to multiplication by a nonzero constant.

The unit circle in $RP^2$ is naturally identified with the lines in $\mathbb{R}^3$ lying in the $xy$-plane, i.e. those $[v_1:v_2:v_3]$ with $v_3=0$. So we want a continuous map from $RP^2\setminus \{*\}$ to points with zero third coordinate. If we send $[v_1:v_2:v_3]$ to $[v_1:v_2:0]$, we certainly have a continuous map, with one problem: if $v_1=v_2=0,$ then we've sent our point to $[0:0:0]$, which are not the coordinates of any point of $RP^2$. That's why we have to rule out a single point, namely $[0:0:1]$.

Elsewhere our map is well defined (as you should check) and it's not hard to see it's homotopic to the identity: the homotopy is $H_t([v_1:v_2:v_3])=[v_1:v_2:tv_3]$, which steadily tilts a line down into the $xy$-plane. On the other hand our map is the identity when restricted to the circle in $RP^2$, so we have a deformation retract and in particular a homotopy equivalence as desired.

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  • $\begingroup$ may I ask what is your $t$? in the notation $H_t$? $\endgroup$
    – user138017
    Apr 2 '15 at 9:13
  • $\begingroup$ $t$ is the parameter in $[0,1]$ required to define a homotopy. $\endgroup$ Apr 2 '15 at 16:45
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Note the CW-complex structure of $RP^2$ is $e^0\cup_{c}e^1\cup_{f}e^2$,where $c$is the constant map and $f$is the antipodal map.If you remove a point from $RP^2$ then upto homotopy the cell structure of $RP^2 - \lbrace x\rbrace$ is exactly $e^0\cup_{c}e^1$.Which is nothing but $S^1.$

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    $\begingroup$ I think a small part of the argument is missing, namely why can you assume that the removed point is in the interior of $e^2$? Otherwise it might not be 100% clear how to recover the same cell structure. $\endgroup$ Apr 2 '15 at 9:53

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