The universal covering manifold has a construction as follows: Fix a base point $p$ of $M$. Two paths $c_i\colon I \rightarrow M$ ($i=0,1$) on $M$ with terminal point $p$ and the same starting point are homotopic if they are connected by a family of paths $c_s: I \rightarrow M$ ($s \in I$ on $M$) with the same starting and terminal points. With the set of all homotopy classes of paths on $M$ and the same terminal points $p$, the universal covering manifold can be constructed. The construction involves with the low dimensional paths $I \rightarrow X$ and lower dimensional homotopies of paths $I \times I \rightarrow X$. It forms a path lifting property. How to apply higher dimensional cubes $I^n$ and homotopies $I^n \times I \rightarrow X$ for the construction and form the lifting of all homotopies (not just path) ?
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$\begingroup$ Please use dollar signs to texify your post. It might also help readability to break it into two or three paragraphs. $\endgroup$– RasmusCommented Nov 15, 2013 at 19:07
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$\begingroup$ @Rasmus I added dollar signs for readability, but I still don't understand the question. $\endgroup$– Jim BelkCommented Nov 15, 2013 at 19:14
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$\begingroup$ Is it sufficient and necessary to construct the covering with cubes instead of paths ? $\endgroup$– TomCommented Nov 15, 2013 at 19:35
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$\begingroup$ Are you asking for a proof that a universal covering of a manifold is a Serre fibration? $\endgroup$– Dan RustCommented Nov 16, 2013 at 15:46
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$\begingroup$ Homotopic lifting property asserts for arbitrary fibrations. What limits the covering homotopy to Serre fibrations ? How about Hurewicz fibrations and Dold fibrations ? $\endgroup$– TomCommented Nov 16, 2013 at 18:14
1 Answer
Another way of putting the standard construction of the universal covers of a space $X$ at various base points $x \in X%$ is to form the fundamental groupoid $\pi_1 X$ consisting of homotopy classes rel end points of paths $X$. There are two function $s,t: \pi_1 X \to X$ giving the beginning and end, or source and target, of such classes of paths. Then the universal cover of $X$ at the point $x \in X$ is $s^{-1}(x)$, also called the star of the groupoid at $x$. Of course one can also use $t^{-1}(x)$ also called the costar.
One problem of trying to use using cubes is that there is no easy generalisation of the fundamental groupoid to higher dimensions. A generalisation using filtered spaces is explained in the book Nonabelian algebraic topology:.... A filtered space is a space $X$ with an increasing sequence $X_r. r \geqslant 0$, of subspaces. An example is the cube $I^n$ with its filtration $I^n_*$ by skeleta $I^n_r$, consisting of all subcubes of dimension $\leqslant r$. The $\rho_n(X_*)$ consists of homotopy classes rel vertices of maps $I^n_* \to X_*$, where the hpmotopies are through such filtered maps. However the proof that this becomes a multiple groupoid is tricky! This generalisation does have many applications, and covering space notions are involved at various points.
There are also generalisations in dimension $2$ for Hausdorff spaces, see this paper and its sequels.
