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I am currently taking a course in algebraic topology and I have to calculate the universal cover of a lot of spaces (I'll call them $X$ from here on). So far I know some tricks to do it: consider $X'$ simply connected and a group $\Gamma$ which acts proper and discontinuously on $X'$ such that $X \cong X'/\Gamma $ (I calculated the universal covers of the Möbius band, the Klein bottle and the torus with this), try to imagine the covering space knowing that if the spaces have "nice" properties $\#F_p = \# \pi_1(X)$ (I calculated the universal cover of $S^2$ with a loop attached with this), $\dots$

I feel that these methods rely a lot on intuition and I would like to use a general method if possible to avoid having trouble when doing the computation due to my lack of "vision". Is there any trick or general method which I can use to do this calculations?

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    $\begingroup$ There is another way, sometimes effictive... which is desribed in hatcher's page 65 . For example if $X$ is union of two subspace $A$ and $B$ for which you know their simple connected covering spaces. Then assembling those copies, you can construct a simple connected space of X. $\endgroup$ Dec 6 '15 at 16:35
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    $\begingroup$ No, there is no general - effective - way to compute universal cover. How $\widetilde{X}$ would look like depends a lot on $X$. There is a way to construct $\widetilde{X}$ from equivalence classes of paths in $X$ based at a particular point $x_0$ in $X$, but that hardly tells you how $\widetilde{X}$ looks like for any $X$, making it ineffective. There are methods to compute universal cover of product spaces, wedge spaces, and spaces with groups acting "nicely". A combination of these techniques generally suffices to construct universal cover of nice spaces. $\endgroup$ Dec 6 '15 at 16:52
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    $\begingroup$ It depends on what you mean by "calculate." This is in some sense at least as hard as computing the fundamental group, and depending on what you mean by "compute" this is an undecidable problem. $\endgroup$ Dec 6 '15 at 17:08
  • $\begingroup$ Related, perhaps even duplicate. $\endgroup$ Feb 20 '16 at 8:27
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Here's an 'algorithm' that may be helpful in the special case when $X$ is a wedge of two topological spaces: http://www.math3ma.com/mathema/2015/12/17/a-recipe-for-the-universal-cover-of-xy

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    $\begingroup$ Although this link may provide an answer to the posted Question, links on the Internet are at risk of becoming inactive. It would be better to provide a quotation of the most essential part of the relevant material there, if not also some explanation in your own words. This will help not only in reconstruction of a link that goes down, but also inform your Readers about their decision to follow a link (or not). $\endgroup$
    – hardmath
    Feb 20 '16 at 2:55
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    $\begingroup$ Thank you for your comment! I'm brand new to Stack Exchange. This is good to know going forward. $\endgroup$
    – user316092
    Feb 21 '16 at 3:37
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    $\begingroup$ An updated link. $\endgroup$ Feb 18 '19 at 18:46

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