Homotopy class [M,BG]

Let $M$ be a smooth manifold and let $BG$ be the classifying space associated to a topological group $G$ .

The notation [M,BG] denotes the set of (unbased) homotopy classes of continuous maps $f\colon M\to BG$ .

Concretely, two maps $f_0,f_1\colon M\to BG$ define the same element of [M,BG] if there exists a continuous homotopy

H\colon M\times [0,1] \to BG,\qquad H(\cdot,0)=f_0,\; H(\cdot,1)=f_1.

Functoriality and bundle classification

If $\phi\colon M'\to M$ is a diffeomorphism, then precomposition induces a bijection

[M,BG] \xrightarrow{\;\cong\;} [M',BG], \quad [f]\mapsto [f\circ \phi].

Under the usual hypotheses of the classifying-space theorem, this set is the target of the classification map sending a principal G-bundle $P\to M$ to the homotopy class of its classifying map.

Examples

Spheres. The based homotopy set satisfies $[S^n,BG]_*=\pi_n(BG)$ . Passing to unbased classes may further identify elements through the action of the fundamental group, so the basepoint should not be suppressed without an additional hypothesis.
Contractible bases. If $M$ is nonempty and contractible and $BG$ is path-connected, then [M,BG] has exactly one element (every map is homotopic to a constant map).
Line bundles. For $G=U(1)$ one has $BG\simeq \mathbb{C}P^\infty$ , and [M,BG] corresponds to isomorphism classes of principal $U(1)$ -bundles (equivalently complex line bundles) over $M$ .