# $\{\gamma \in C^0([0,1],M): \gamma (0)=p, \gamma (1)=q\}$ with the compact-open top has the homotopy type of a CW complex

...where $M$ is a smooth manifold and $p, q \in M$. Does anyone know of any slick or accessible proofs of this? I was referred to Milnor's "On Spaces Having the Homotopy Type of a CW Complex" which is a bit over my head. I was wondering if anything simpler had been discovered in the intervening decades.

Edit: It looks like these are the steps Milnor takes (for proofs of which he cites papers I can't find or papers in languages i don't read):

Lemma 1: Let $X$ be a topological space. $X$ has the homotopy type of an absolute neighborhood retract (ANR) iff $X$ has the homotopy type of a countable CW complex

Lemma 2: If $Y$ is compact metric and $X$ is an ANR then the space of maps $Y\rightarrow X$ is an ANR.

From these lemmas I can prove that $\{\gamma \in C^0([0,1],M)\}$ has the homotopy type of a CW-complex since I can prove $M$ has the homotopy type of a countable CW-complex. But I don't know how to get from there to the homotopy type of a subspace where the endpoints are fixed. And I don't know how to prove the lemmas.

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## 1 Answer

The basic idea is to put a complete Riemann metric on $M$ and to look at the subspace of your space of paths that consist of paths that are piecewise geodesic. So there is no restriction on the speed of the parametrization, these paths can go from any speed to any other speed (in pieces), including constant pieces. Alternatively, if $M$ isn't a manifold but just a simplicial complex, you could look at the subspace of your path space where the paths are piecewise linear. These spaces have natural filtrations by the minimal number of subdivisions you need in the "piecewise" description of the path. Of course, this is just a stratified object, not quite a CW-complex. You have to do more work to get the actual result but the above is a pencil-sketch idea you can try to flesh out...

If you would like to see another take on Milnor's construction, try this paper of Bahri and Cohen: http://front.math.ucdavis.edu/0806.0637

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Thanks Ryan! The question comes from my reading "Morse Theory" in which Milnor proves that this path space is h-equiv to the space of piecewise smooth curves (w the same condition on the endpnts). In turn, the space of such piecewise smooth curves with energy smaller than a given constant is h-equiv to a space of broken geodesics with the same energy bound.Finally, this latter space has a smooth manifold structure, and is therefore h-equiv to a cw cmplx. If I could remove the energy bound I would have the result in the title. Is this something like what youre suggesting? – Tim kinsella Dec 13 '11 at 10:53
These piecewise geodesic subspaces give you an inclusion, so to get the homotopy-equivalence you need a map back. The problem with constructing a map back is you need to choose the amount of subdivision. I don't have Milnor's book in front of me, but presumably you can use this "energy" to pick a subdivision, so that you can simply choose appropriate points along the curve and take geodesics between them (the points would presumably be within the injectivity radius of the manifold). – Ryan Budney Dec 13 '11 at 19:52
Ok thank you. I will try to work through the Bahri and Cohen paper. Is there a particular result in the paper that implies the statement? – Tim kinsella Dec 15 '11 at 8:11