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I'm tearing my hair out trying to prove that a contractible subcomplex ,$K$, of a contractible CW complex, $L$, is a strong deformation retract of $L$.  

What I have so far:

  1. I can show that the contractibility of K gives a retract $ L \rightarrow K$.

  2. I also think I can show that K is a (not necessarily strong) deformation retract of L.  

    2.1 I also know it to be a fact that a subcomplex is a def retract iff it's also a strong def retract but I dont know how to prove this.

3.Lastly I've also reduced the problem to the case where $K$ is a point as follows:    since $K$ is a subcomplex, a def retract of $K$ onto $p\in K$ extends to a homotopy of $L$.  Then if the statement held for K a point, we could perform a subsequent strong def retract onto $p$,  furnishing a contraction of $L$ which maps $K$ to itself at each stage of the homotopy.  This provides a contraction of the space $K\times I \cup L\times 0 \cup L\times 1$. But this space is a subcomplex of the complex $L\times I$ and so $L\times I$ retracts onto$K\times I \cup L\times 0 \cup L\times 1$.  The statement follows.

 So yeah I'd be very grateful for any hints or solutions.  Thanks for your time.

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Have you looked at the proof of the Whitehead theorem? – Ryan Budney Feb 23 '12 at 10:38
That does it! Thanks Ryan. – Gary soto Feb 23 '12 at 11:24

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