Integral Cohomology and Geometry

Geometric Realization of Second Integral Cohomology Classes

The second integral cohomology classes in \( H^{2}(M ; Z) \) are in natural \( 1-1 \) correspondance with the isomorphism classes of circle-bundles over \( M \), where $M$ is a manifold.

In particular, every second integral cohomology class \( \hat{F} \in H^{2}(M ; Z) \) has a geometric realization which consists of a space \( P=P_{F} \) together with an action of the circle \( S^{1} \) on \(P \) and a natural projection:

\[\begin{array} \\ P \\ \downarrow \pi \\ M \end{array}\]

so that the \( S^{1} \) action on \( P \) is free, preserves \( \pi, \) and \( \pi \) induces an isomorphism of \( P / S^{1} \) with \( M .\)

For instance if \( M \) is a Riemann surface, consider that $P$ is the set of its unit tangent vectors to $M$, and let \( \pi \) be the projection map which assigns to each tangent vector the corresponding point of tangency. The circle \( S^{1} \) then acts freely by rotating any vector counter-clockwise in its tangent plane. The simplest example of a \( P \) is \( M \times S^{1} \) with \( S^{1} \) acting on itself by right translation. This \( P \) is called the trivial bundle and it corresponds to the neutral element $0$ in \( H^{2}(M ; Z) .\) Locally all bundles \( P \) are isomorphic to such a trivial one, but globally \( P \) will in general not equal \( M \times S^{1} . \) Consider the unit tangent bundle \( P \) of the 2-sphere \( S^{2}, \) being isomorphic to the group of rotations of 3-space \( S O(3) \), and this \( P \) is not trivial. Actually, it is the geometric realization of twice the generator of:

$$ H^{2}(M ; Z) \simeq Z \quad  \quad M=S^{2}$$ The double covering of \( S O(3) \) is the 3-sphere \( S^{3} \) and the resulting natural map \( S^{3} \rightarrow S^{2} \) is the Hopf fibration, which geometrically represents the generator of \( H^{2}(M ; Z) .\)