fibre bundle post

_posts/2024-05-01-fiber-bundles-and-cech-cohomology.md

@@ -0,0 +1,45 @@
+---
+layout: post
+title: Fiber Bundles and Čech cohomology
+---
+{% include mathjax.html %}
+
+# An analysis of the Möbius band
+
+The [Möbius band](https://en.wikipedia.org/wiki/M%C3%B6bius_strip) is a simple example of a nontrivial vector bundle (or more generally, a [fiber bundle](https://en.wikipedia.org/wiki/Fiber_bundle_construction_theorem).)
+
+![A Möbius band](/blog/docs/assets/images/moebius.png)
+
+If anyone was asked to describe what is special about the Möbius band, they would point out the single twist.
+
+![A Möbius band with the twist highlighted](/blog/docs/assets/images/moebius-twist.png)
+
+Indeed that is what characterizes the Möbius band, as nothing else happens; and yet mathematically we'd also have to point out that the rest is boring, as "a twist" is not a full description (of course in regular communication it implies that the rest is "as-is", i.e. without a twist.)
+
+![A Möbius band with the twist and regular part highlighted](/blog/docs/assets/images/moebius-twist2.png)
+
+The description of the Möbius band we arrive at says that on one area, there's a twist, and on the other area, there's no twist. How large should the circled areas be? It is obvious that making these areas a bit smaller or larger would not affect our description. How wide should the Möbius band be? It is again obvious that making the band narrower would not affect its character. By taking these ideas to their limit, we arrive at this distilled description of the Möbius band:
+
+![A limit description of the Möbius band](/blog/docs/assets/images/moebius-limit.svg)
+
+The boring part is labeled $1$ while the twist is $-1$. We cannot reduce the description further. The meaning of $1$ and $-1$ comes from the description of the transition maps for a 1-dimensional vector bundle, namely they are elements of the general linear group $\operatorname{GL}(1, \mathbb{R})$ which is group-isomorphic to the multiplicative group $\mathbb{R}\setminus 0$. If we name the two circled areas of the Möbius band as $V$ and $W$, then we have maps $V, W\to\operatorname{GL}(1,\mathbb{R})$. Just as we reduce $V$ and $W$ to points their images also reduce to values on $\operatorname{GL}(1,\mathbb{R})$, and it is immaterial to which as long as the component is respected; $\operatorname{GL}(1,\mathbb{R})$ has two components, $\mathbb{R}_-$ and $\mathbb{R}_+$.
+
+## What is this distillation process?
+
+If we repeat the analysis of the Möbius band for the trivial bundle $S^1\times\mathbb{R}$ (a cylinder), we will end up with $S^1$ and a single point with the value $1$.
+
+![A limit description of the trivial bundle over the circle](/blog/docs/assets/images/cylinder-limit.svg)
+
+Can we reverse this process to obtain the original bundles from their limit pictures? The [Fiber Bundle Construction Theorem](https://en.wikipedia.org/wiki/Fiber_bundle_construction_theorem) tells us that we can construct fiber bundles from the transition maps $U_i\cap U_j \to G$ for an [open cover](https://en.wikipedia.org/wiki/Cover_(topology)) $U_i$ of the base. This is very close to what we had just before taking limits in the Möbius band; the areas $V$ and $W$ are the intersections of the cover $U_1, U_2$ of $S^1$.
+
+This already gives us intuition for the Fiber Bundle Construction theorem, but there is another connection we can make! The maps $U_i\cap U_j \to G$ also describe a $G$-valued [Čech 1-cochain](https://en.wikipedia.org/wiki/%C4%8Cech_cohomology). The limit description of the Möbius band is nothing but an element of the first Čech cohomology group $H^1(S^1, \operatorname{GL}(1, \mathbb{R}))$.
+
+## Yet another limit
+
+It is possible to take yet another limit; so far we've only shrunk down the fibers, but conceivably the base space $B$ could also be shrunk. For instance, had the base space $S^1$ been widened to $S^1\times(-1,1)$ for example, the 1-dimensional vector bundle $(S^1\times(-1,1))\times\mathbb{R}$ to be distilled to the same picture as the cylinder. On a Mathematical level, since we established a construction of fiber bundles via $H^1(B, G)$, we are simply leveraging the fact that cohomology is preserved for homotopy-equivalent spaces $B$ and $B'$.
+
+## Notes
+
+1. There is a wonderful post by Zhen Lin on stackexchange which explains the isomorphisms for Čech cohomology and fiber bundle constructions, <https://math.stackexchange.com/a/197989>.
+2. The reader should be **warned** that Čech cohomology is not typically discussed for nonabelian groups, because it does not go beyond the first cohomology group (as we've seen, it completely classifies the fiber bundle space.) Čech cohomology is almost always in the context of abelian groups.
+3. The Möbius band was created using [sage's](https://www.sagemath.org/) `sage.plot.plot3d.parametric_surface.MoebiusStrip`, color corrected with [GIMP](https://www.gimp.org/), and cropped via [ImageMagick](https://imagemagick.org/index.php). The limit process pictures were created in [Inkscape](https://inkscape.org/).