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I'm trying to compute characteristic classes of principal bundles by defining transition functions and computing them in Čech cohomology. However, it seems all the constructions are defined in terms of "good open covers" (where the sets in the cover and all finite intersections are contractible), and in practice it seems very difficult to actually work with a good open cover (for example, it seems the smallest good open cover of $RP^2$ has 10 open sets, and things quickly get worse for more complicated manifolds). Instead, one can try to decompose the manifold into simpler pieces, and "glue" these together. My question is vague, but is there some general way to think about such things?

As a very simple example, say I want to compute the second Stiefel-Whitney class, $w_2$, of an $\newcommand{\SO}{\mathrm{SO}} \SO(N)$ bundle over $S^2$. In this case I can find a simple good open cover, say, take a tetrahedron and open sets which are an epsilon neighborhood of each face. Then the Čech cohomology computation of $w_2$ goes as follows: the bundle is determined by the $\SO(N)$ valued transition functions on each double-overlap, which are essentially functions $g_{ij}:I\to\SO(N)$, where $i,j \in \{1,2,3,4\}$, which lift uniquely (up to an overall sign) to $\newcommand{\Spin}{\mathrm{Spin}} \hat{g}_{ij}:I\to\Spin(N)$. Then, on triple overlaps, the vertices, these must satisfy $\hat{g}_{ij} \hat{g}_{jk} \hat{g}_{ki} = \epsilon_{ijk}$, a sign which determines the Čech 2-cocycle with coefficients in $Z/2$ giving the Stiefel-Whitney class, and whose contraction with the fundamental class is just the product of the four signs.

Alternatively, one can save a little time here (and much more in more complicated spaces) by just decomposing the sphere into two hemispheres, and then gluing along the equator by a function $g:S^1\to\SO(N)$. Then this overlap is not contractible, but we can cover it by a single interval, which overlaps itself slightly, and on this interval the path $g$ lifts to a path $\hat{g}:I\to\Spin(N)$, such that at the overlap there is a mismatch by a sign, which is just the Stiefel-Whitney class (or more precisely, its contraction with the fundamental class).

With some thought one can see why these are equivalent in this case, and I can reproduce similar kinds of arguments on slightly more complicated spaces, but I'm having a hard time seeing the bigger picture behind what I'm doing. For example, what is the "shortcut" calculation for $RP^2$? Is there a more generalized form of Čech cohomology that works even for covers which are not good, or a general method for gluing subspaces along topologically non-trivial intersections? Or maybe there's a completely different, better way to compute the kinds of things I'm after.

Edit:

To make the question a little more concrete - suppose I am given a principal bundle over $X$, and $X$ decomposes into a union of open sets $A$ and $B$. Then is there a nice way to express the characteristic classes of the bundle on $X$ in terms of those of the restriction of the bundle to $A$, $B$, and $A \cap B$ (and whatever other data is necessary)?

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    $\begingroup$ The keyword should be "homotopy colimits." Ordinary Čech theory is the special case where you consider homotopy colimits of points. $\endgroup$ Dec 7, 2013 at 18:30
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    $\begingroup$ Another keyword is "hypercover". $\endgroup$
    – S. Carnahan
    Dec 8, 2013 at 0:52
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    $\begingroup$ Another keyword is "sheaf". You want to glue things, so you need a sheaf. Cohomology is not a sheaf. For degree 1 cohomology, you can use principal bundles: these form a sheaf. For degree 2 cohomology, i.e. for $w_2$, you need gerbes: these form a 2-sheaf, or stack. $\endgroup$ Dec 8, 2013 at 9:16
  • $\begingroup$ @Konrad Waldorf Do you happen to know of a reference where I could learn more about this (gerbes, stacks, and Cech cohomology)? $\endgroup$
    – ಠ_ಠ
    Sep 2, 2016 at 6:46

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