When dealing with some lifting problems, I came across the following problem, which probably has a well-known answer, but anyway:

Suppose I have a (locally contractible) topological group $G$, such that $\pi_1(G) \cong \mathbb{Z}$. Let $\tilde{G} \to G$ be its universal covering group. Let $P \to X$ be a principal $G$-bundle over a compact Hausdorff space $X$. The obstruction of lifting $P$ to a principal $\tilde{G}$-bundle lives in $H^2(X, \pi_1(G)) \cong H^2(X, \mathbb{Z})$ and is therefore given by a complex line bundle over $X$.

Is it possible to construct this line bundle directly from $P$ (without going through the classification of line bundles by their Chern classes)?

When dealing with some lifting problems, I came across the following problem, which probably has a well-known answer, but anyway:

Suppose I have a (locally contractible) connected topological group $G$, such that $\pi_1(G) \cong \mathbb{Z}$. Let $\tilde{G} \to G$ be its universal covering group. Let $P \to X$ be a principal $G$-bundle over a compact Hausdorff space $X$. The obstruction of lifting $P$ to a principal $\tilde{G}$-bundle lives in $H^2(X, \pi_1(G)) \cong H^2(X, \mathbb{Z})$ and is therefore given by a complex line bundle over $X$.

Is it possible to construct this line bundle directly from $P$ (without going through the classification of line bundles by their Chern classes)?

When dealing with some lifting problems, I can across the following problem, which probably has a well-known answer, but anyway:

Suppose I have a (locally contractible) topological group $G$, such that $\pi_1(G) \cong \mathbb{Z}$. Let $\tilde{G} \to G$ be its universal covering group. Let $P \to X$ be a principal $G$-bundle over a compact Hausdorff space $X$. The obstruction of lifting $P$ to a principal $\tilde{G}$-bundle lives in $H^2(X, \pi_1(G)) \cong H^2(X, \mathbb{Z})$ and is therefore given by a complex line bundle over $X$.

Is it possible to construct this line bundle directly from $P$ (without going through the classification of line bundles by their Chern classes)?

When dealing with some lifting problems, I came across the following problem, which probably has a well-known answer, but anyway:

Suppose I have a (locally contractible) topological group $G$, such that $\pi_1(G) \cong \mathbb{Z}$. Let $\tilde{G} \to G$ be its universal covering group. Let $P \to X$ be a principal $G$-bundle over a compact Hausdorff space $X$. The obstruction of lifting $P$ to a principal $\tilde{G}$-bundle lives in $H^2(X, \pi_1(G)) \cong H^2(X, \mathbb{Z})$ and is therefore given by a complex line bundle over $X$.

Is it possible to construct this line bundle directly from $P$ (without going through the classification of line bundles by their Chern classes)?