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The question here asks for examples of deformation retracts which are not strong deformation retracts. A comment is given refering the asker to Exercise 0.6 in Hatcher. (The description of this space also appears in this question, which is about this very same exercise.)

This space, however, seems to be constructed specifically for this purpose.

Are there any naturally occurring (by which I mean not constructed specifically for this purpose, arising in some other way) spaces that are deformation retracts but not strong deformation retracts?

The main reason for asking is that it doesn't seem very likely to me that one would stumble upon such a space at random, so I don't know what to watch out for. One can never be too careful.

Community
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Dejan Govc
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2 Answers2

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This depends on what you mean by being "specifically constructed for this purpose". A nice example (that I'm sure inspired the example given by Hatcher) is the comb space $$X=[0,1]\times\{0\}\cup\{0\}\times [0,1]\cup\bigcup_{n=1}^\infty \left(\left\{\frac{1}{n}\right\}\times [0,1]\right)$$

Any connected subset of the leftmost tooth $\{0\}\times[0,1]$ is a deformation retract of $X$, since $X$ contracts to the point $(0,0)$. But most of the time these won't be strong deformation retracts; in fact the only subset of that tooth that is a strong deformation retract is the point $(0,0)$.

Miha Habič
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    Thanks, this is a perfect answer to the question I asked, so I've decided to accept it. I see I've asked the wrong question, though. =/ What I'm actually interested in is nicer examples of spaces that deformation retract to some point but do not strongly deformation retract to any point. I guess I'll do some more thinking and ask a separate question in case I fail to reach a conclusion. – Dejan Govc Mar 30 '13 at 16:28
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For completeness, I think the answer to this question is answered in Spanier's Algebraic Topology in Theorem 1.4.11 on p.31:

If $(X \times I, (X \times 0) \cup (A \times I) \cup (X \times 1))$ has the homotopy extension property with respect to $X$ and $A$ is closed in $X$, then $A$ is a deformation retract of $X$ if and only if $A$ is a strong deformation retract of $X$.

mcat
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