We present the results of studying free massive bosonic fields under the formalism of topological quantization. We identify certain harmonic maps as a geometric configuration equivalent to the classical system. We take as a concrete example the case of free massive bosonic fields in two dimensions, and construct the fiber bundle representing them and find its topological spectra. We found that the appearance of singularities in Euler's two form caused its integral to be dependent on the order in which the variables are integrated. We discuss the implications of this orientation dependency and formulate a well-defined expression for the Euler invariant emerging from it.