Defining Pontrjagin numbers

Question

I'm reading about Pontrjagin numbers and I have difficulties understanding how they are defined. I have the following definition.

Let $E$ be a vector bundle of rank $r$ over a compact manifold $M$ of dimension $4m$. A monomial $p^{a_1}_1p^{a_2}_2\cdots p^{a_{\lfloor r/2\rfloor}}_{\lfloor r/2\rfloor}$ of weighted degree $$4\left(a_1+2a_2+\dots+ \left\lfloor \frac{r}{2}\right\rfloor a_{\left\lfloor r/2\right\rfloor}\right)=4m$$ represents a cohomology class of degree $4m$ on $M$ an can be integrated over $M$, the resulting number $\int_M p^{a_1}_1p^{a_2}_2\cdots p^{a_{\lfloor r/2\rfloor}}_{\lfloor r/2\rfloor}$ is called a Pontrjagin number of $E$.

Here $p_k=f_{2k}\left(\frac{i}{2\pi}\Omega\right)$ where $f_{2k}$ are the generators of the space of invariant polynomials and $\Omega$ the curvature matrix.

Could someone here elaborate on why do we need to take the floor of $r/2$? I understand that it is due to the forms vanishing of some kind of dimension reasons?

Also what is this consideration of a monomial $p^{a_1}_1p^{a_2}_2\cdots p^{a_{\lfloor r/2\rfloor}}_{\lfloor r/2\rfloor}$ of weighted degree $4\left(a_1+2a_2+\dots+ \left\lfloor \frac{r}{2}\right\rfloor a_{\left\lfloor r/2\right\rfloor}\right)=4m$, I'm not grasping the idea here?

Answer 1

For $x \in \mathfrak{gl}_r(\mathbb{R})$, we have $\det\left(I-t\dfrac{x}{2\pi}\right) = \displaystyle\sum_{k=0}^rf_k(x)t^k$ and $p_k(E) = [f_{2k}(\Omega)]$ where $\Omega$ is the curvature of a connection on $E$. So for each even integer $2i \leq r$, there is a corresponding Pontryagin class $p_i(E) = [f_{2i}(\Omega)]$. Note that the largest even integer less than or equal to $r$ is $2\lfloor r/2\rfloor$ (which is $r$ itself if $r$ is even, or $r - 1$ if $r$ is odd), so $E$ has Pontryagin classes $p_1(E), p_2(E), \dots, p_{\lfloor r/2\rfloor}(E)$.

Since $\Omega$ is a $\mathfrak{gl}_r(\mathbb{R})$-valued $2$-form, we see that $f_{2k}(\Omega)$ is a $4k$-form, and hence $p_k(E) = [f_{2k}(\Omega)] \in H^{4k}_{\text{dR}}(M)$. So $p_k(E)^{a_k}$ has degree $4ka_k$, and hence $p_1(E)^{a_1}p_2(E)^{a_2}\dots p_{\lfloor r/2\rfloor}(E)^{a_{\lfloor r/2\rfloor}}$ has degree

$$4a_1 + 8a_2 + \dots + 4\left\lfloor\frac{r}{2}\right\rfloor a_{\lfloor r/2\rfloor} = 4\left(a_1 + 2a_2 + \dots + \left\lfloor\frac{r}{2}\right\rfloor a_{\lfloor r/2\rfloor}\right).$$

When this degree is equal to $4m = \dim M$, then we can calculate the integral $\displaystyle\int_Mp_1(E)^{a_1}p_2(E)^{a_2}\dots p_{\lfloor r/2\rfloor}(E)^{a_{\lfloor r/2\rfloor}}$.