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 Post subject: Fixed point?
PostPosted: Thu, 6 Oct 2011 00:47:59 UTC 
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This one I don't have any clue how to do, a friend asked it:

If X,Y are topological spaces such that any self-maps have a fixed point, does this hold for X\wedge Y? If so why?

Edit: Oops, this should be the V wedge, the one where we have the coproduct and glue them at a single point.

Edit 2: bolded Edit 1

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 Post subject: Re: Fixed point?
PostPosted: Thu, 6 Oct 2011 01:08:58 UTC 
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Shadow wrote:
This one I don't have any clue how to do, a friend asked it:

If X,Y are topological spaces such that any self-maps have a fixed point, does this hold for X\wedge Y? If so why?

Edit: Oops, this should be the V wedge, the one where we have the coproduct and glue them at a single point.

Edit 2: bolded Edit 1


Lemma: If X,Y have the fixed point property (fpp), then the one-point union X\vee Y has fpp too.
Proof: Let x\in X,y\in Y with X\vee Y=(X\amalg Y)/(x=y), and let p be the common point {x,y}. Let f\colon X\vee Y\to X\vee Y. Assuming f(p)\neq p, so WLOG f(p)\in X. Now r\colon X\vee Y\to X defined by r\vert_X=\mathrm{id}_X, r(Y)=p is a retract of X\vee Y to X. So r\circ f\vert_X\colon X\to X is continuous, hence fpp of X tells us f has a fixed point.

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\begin{aligned}
Spin(1)&=O(1)=\mathbb{Z}/2&\quad&\text{and}\\
Spin(2)&=U(1)=SO(2)&&\text{are obvious}\\
Spin(3)&=Sp(1)=SU(2)&&\text{by }q\mapsto(\mathop{\mathrm{Im}}\mathbb{H}\ni p\mapsto qp\bar{q})\\
Spin(4)&=Sp(1)\times Sp(1)&&\text{by }(q_1,q_2)\mapsto(\mathbb{H}\ni p\mapsto q_1p\bar{q_2})\\
Spin(5)&=Sp(2)&&\text{by }\mathbb{HP}^1\cong S^4_{round}\hookrightarrow\mathbb{R}^5\\
Spin(6)&=SU(4)&&\text{by the irrep }\Lambda_+\mathbb{C}^4
\end{aligned}


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 Post subject: Re: Fixed point?
PostPosted: Thu, 6 Oct 2011 01:14:47 UTC 
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outermeasure wrote:
Shadow wrote:
This one I don't have any clue how to do, a friend asked it:

If X,Y are topological spaces such that any self-maps have a fixed point, does this hold for X\wedge Y? If so why?

Edit: Oops, this should be the V wedge, the one where we have the coproduct and glue them at a single point.

Edit 2: bolded Edit 1


Lemma: If X,Y have the fixed point property (fpp), then the one-point union X\vee Y has fpp too.
Proof: Let x\in X,y\in Y with X\vee Y=(X\amalg Y)/(x=y), and let p be the common point {x,y}. Let f\colon X\vee Y\to X\vee Y. Assuming f(p)\neq p, so WLOG f(p)\in X. Now r\colon X\vee Y\to X defined by r\vert_X=\mathrm{id}_X, r(Y)=p is a retract of X\vee Y to X. So r\circ f\vert_X\colon X\to X is continuous, hence fpp of X tells us f has a fixed point.


But what if f|_{_X} has range in Y?

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 Post subject: Re: Fixed point?
PostPosted: Thu, 6 Oct 2011 01:15:57 UTC 
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Oh, nvm, that's why r is there, to fix that.

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