About the homotopy type of the path space My sole question is: Given two homotopy equivalent topological spaces $X$ and $Y$ and points $p \in X$, $q\in Y$, then what do we know about the two associated loop spaces $\Omega(X;p,p)$ and $\Omega(Y;q,q)$ (which consists of continuous loops $\gamma:p\leadsto p$ in $X$ resp. $\gamma: q\leadsto q$ in $Y$) given the compact-open-topology.
Are they homotopy equivalent aswell? If so, how do I see that?
My guess is that I need $X$ and $Y$ to be pointed homotopy equivalent to conclude that the loop spaces are homotopy equivalent aswell.
(I'm questioning this as I believe Milnor is arguing that way in his Morse Theory book, corollary 17.5, p.96).
 A: It depends. In general they can have nothing in common. However, if $X$ and $Y$ are path-connected, or more generally if there is a homotopy equivalence $X\to Y$ sending $p$ to the connected component of $q$, then they are indeed homotopy equivalent. 
The first thing to see is that $\Omega(X,p)$ only depends (up to homotopy) on the path-connected component of $p$. 
Indeed, let $\gamma$ be a path from $p$ to $q$ in $X$, and define $\beta_\gamma: \Omega(X,p)\to \Omega(X,q)$ by $\delta\mapsto \overline\gamma \delta\gamma$
Now there are three interesting properties : the first is that if $\gamma$ is path-homotopic to $\gamma'$ then $\beta_{\gamma}$ and $\beta_{\gamma'}$ are homotopic. Indeed, let $H$ be a homotopy from $\gamma$ to $\gamma'$ and let $K(\delta,s) = \overline{H(-,s)}\delta H(-,s)$ where $H(-,s)$ denotes the path $t\mapsto H(t,s)$. One easily checks that this is continuous, and clearly for $s=0$ or $s=1$ we find $\beta_\gamma$ or $\beta_{\gamma'}$. 
The second is that if $\gamma$ is a path from $p$ to $q$ and $\gamma'$ a path from $q$ to $r$, then $\beta_{\gamma'}\circ \beta_\gamma$ is homotopic to $\beta_{\gamma\gamma'}$ (notice the inversion, but that's just because of the conventions to denote path concatenation). This is again quite easy to check, with a proof very similar to the one above where you just have to reparametrize. 
The third is that if $\gamma$ is the constant path at $p$, then $\beta_\gamma$ is homotopic to the identity : again this is fairly straightforward, and just a matter of reparametrization of the interval. 
All these together say essentially that $p\mapsto \Omega(X,p), \gamma\mapsto \beta_\gamma$ is a functor from the fundamental groupoid of $X$ to the category of spaces modulo homotopy (actually you can extract a better statement from the proof but let's not bother with that). If you know what this means, then good, you know that this automatically implies that if $p,q$ are connected by a path, then $\Omega(X,p)$ is homotopy equivalent to $\Omega(X,q)$. 
If you don't know what these fancy words mean, no worries, here's a sketch of the conclusion  : if $\gamma$ is a path from $p$ to $q$, then we get $\beta_\gamma : \Omega(X,p)\to \Omega(X,q)$, $\beta_{\overline{\gamma}} : \Omega(X,q) \to \Omega(X,p)$. When you compose them, you get a homotopy to $\beta_{\gamma\overline{\gamma}}$ or the other way around by the second property, and then to $\beta_{cstpath}$ by the first property, and so to $id$ by the third property: they are homotopy equivalences. 
Good, now we've cleared this whole "path components business". Now to prove my first claim we only have to show that if $f:X\to Y$ is a homotopy equivalence, then so is $f_*:\Omega(X,p)\to \Omega(Y,f(p))$, where $f_*(\gamma) = f\circ \gamma$. You may check that this is indeed continuous. 
The problem is we have to deal with basepoints. First, you can use some sort of 2-out-of-6 property to reduce to the claim that "if $f:X\to X$ is homotopic to the identity, then $f_*:\Omega(X,p)\to \Omega(X,f(p))$ is a homotopy equivalence". If you don't see how to reduce to that claim, I'll add a few words about it but for now I'll take that as granted and just prove said claim. 
But now say we have a homotopy $H$ from the identity to $f$, let $\gamma$ denote the path $t\mapsto H(p,t)$, and $\gamma_s : t\mapsto H(p,ts)$
Then consider $\beta_\gamma\circ f_*(\delta) = \overline{\gamma}(f\circ \delta)\gamma = \overline{\gamma}(H(-,1)\circ \delta)\gamma$. This gives an idea for a homotopy to $id$ : put $K(\delta,s) = \overline{\gamma_s}(H(-,s)\circ \delta)\gamma_s$ (check that it makes sense and is continuous !). 
Then at $s=1$ this is obviously our $\beta_\gamma\circ f_*(\delta)$, and at $s=0$ this is $\beta_{cstpath}$ which is known to be homotopic to the identity by what we did earlier. Therefore $\beta_\gamma\circ f_*$ is homotopic to the identity. 
Moreover, $f_*\circ\beta_\gamma = \beta_{f\circ\gamma}\circ f_*$, and $f\circ \gamma$ is path homotopic to $H(f(p),-)$ (this is an instance of the Eckmann-Hilton argument: I can expand on this as well if you'd like, though I recommand you draw it and see what it's all about. The Eckmann-Hilton argument actually provides a homotopy from $\gamma (f\circ \gamma)$ to $\gamma H(f(p),-)$, but you can of course remove the $\gamma$'s)  so that the above argument applies and shows that this is, too, homotopic to the identity. 
The result now follows
ADDENDUM : proof that $\gamma (f\circ \gamma)$ is homotopic to $\gamma H(f(p),-)$ (from which the claimed homotopy follows, by simplifying by $\gamma$). 
Consider the two following paths in $X\times [0,1]$ : the first one, $c_1$ goes from $(p,0)$ to $(p,1)$ linearly along the second coordinate, then from $(p,1)$ to $(f(p),1)$ following $\gamma$ on the first coordinate. 
The second one, $c_2$ goes from $(p,0)$ to $(f(p),0)$ following $\gamma$ on the first coordinate, then from $(f(p),0)$ to $(f(p),1)$ linearly along the second coordinate. 
It is easy to check that $c_1\sim c_2$ (define a homotopy coordinate-wise : if you fix a coordinate it's just the fact that they are each of the form $cst\cdot e$ and $e\cdot cst$ for some path $e$). 
Apply the homotopy $H$ to these paths : $H\circ c_1\sim H\circ s_2$ for obvious reasons. But now one checks that $H\circ c_1$ is $\gamma\cdot (f\circ \gamma)$, and $H\circ c_2$ is $\gamma\cdot H(f(p),-)$. This was our claim.
Note : in this addendum I've essentially reproved in a special case (which obfuscates the large picture) that a homotopy between continuous maps induces a natural transformation between the induced functors on fundamental groupoids; then used that natural isomorphisms $id\implies f$ usually satisfy additional properties, which can be seen through Eckmann-Hilton-like arguments - this is what I was talking about in the comments
