The number of linearly independent vector fields on $S^7\times S^5$ The product of the 7-sphere and the 5-sphere: $S^7 \times S^5$, is parallelizable. But how many linearly independent vector fields does it have, and how is this calculated? What books/references would answer this?
 A: A collection of vector fields $\{V_1, \dots, V_k\}$ on a manifold $M$ are said to be linearly independent if $\{V_1(p), \dots, V_k(p)\} \subset T_pM$ is linearly independent for every $p \in M$.
Let $i(M)$ denote the maximal number of linearly independent vector fields on $M$, known as the span of $M$, then $0 \leq i(M) \leq \dim M$. Note that a single vector field is linearly independent if and only if it is nowhere-zero, so $i(M) = 0$ if and only if $M$ does not admit a nowhere-zero vector field, which is equivalent to $\chi(M) \neq 0$ for $M$ closed. Whereas $i(M) = \dim M$ if and only if the tangent bundle is trivial (compare with this question), i.e. $M$ is parallelisable
In particular, as $S^7\times S^5$ is parallelisable, we have $i(S^7\times S^5) = \dim(S^7\times S^5) = 12$.
In general, the calculation of $i(M)$ is a very difficult problem. For example, Adams showed in Vector Fields on Spheres that $i(S^{n-1}) = \rho(n) - 1$. Here $\rho(n)$ is the $n^{\text{th}}$ Radon-Hurwitz number which is determined as follows: if $n = 2^{4a + b}u$ where $a$ is a non-negative integer, $b \in \{0, 1, 2, 3\}$, and $u$ is an odd integer, then $\rho(n) = 8a + 2^b$. So for example,
$$i(S^{13}) = \rho(14) - 1 = \rho(2^{4(0) + 1}\times 7) - 1 = 8(0) + 2^1 - 1 = 1$$
while
$$i(S^{15}) = \rho(16) - 1 = \rho(2^{4(1)+0}\times 1) - 1 = 8(1) + 2^0 - 1 = 8.$$
A: For the sake of completeness: Every finite product of spheres 
$$
S^{n_1}\times S^{n_2}\times ... \times S^{n_k},
$$
$k\ge 2$ and $\forall n_i\ge 1$, one of which is odd-dimensional, is parallelizable, including, of course, $S^5\times S^7$. This is proven by Kervaire in 
M. Kervaire, Courbure integrate generalisee el homotopie, Math. Ann. 131 (1956), p. 219-252 
See also here:
E.B. Staples, A short and elementary proof that a product of spheres is parallelizable if one of them is odd. Proc. Am. Math. Soc. 18, 570–571 (1967) p. 80.
for a short proof. 
