Relationship between $H^1(X, \mathbb{T})$ and complex line bundles

Question

Let $X$ be a compact metric space and consider the sheaf cohomology group $H^1(X, \mathbb{T})$. From a class in $H^1(X, \mathbb{T})$, I can get a principal $\mathbb{T}$-bundle over $X$ and from this, an associated complex line bundle. What is the relationship between classes in $H^1(X, \mathbb{T})$ and complex line bundles? Does one class give me the same line bundle up to isomorphism?

Answer 1

At least if $X$ has the homotopy type of a CW complex, there is a natural isomorphism between $H^1(X; \mathbb T)$ and the group of isomorphism classes of line bundles on $X$ under tensor product.

The usual way this is phrased is that the first Chern class defines an isomorphism from the group of line bundles to $H^2(X;\mathbb Z)$. For example, and for a proof, see Hatcher, "Vector bundles and $K$-theory," Prop. 3.10 (p. 86).

Now consider the short exact sequence of sheaves

$$0\to \mathbb Z\to\mathbb R\to\mathbb R/\mathbb Z\to 0,$$

where $\mathbb R$ carries the continuous topology (i.e., this is the sheaf of continuous real-valued functions on $X$). We have $\mathbb R/\mathbb Z\cong\mathbb T$. There is an induced long exact sequence in cohomology, but as Donu Arapura notes in an answer to a different MathOverflow question, $H^k(X;\mathbb R)$ vanishes for $k > 0$. Therefore the long exact sequence simplifies to

$$ 0 \to H^1(X; \mathbb T)\longrightarrow H^2(X; \mathbb Z)\to 0, $$

so $H^1(X;\mathbb T)$ is isomorphic to the group of line bundles. It takes a little more work to see that the isomorphism is the same as the map you described (associated line bundle to a principal $\mathbb T$-bundle), but that is also true.

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Not all compact metric spaces have the homotopy type of CW complexes, as noted by Milnor (end of section 1). I unfortunately don't know what the answer to your question is for those spaces.