The conjecture on odd-degree torsion in the cohomology of M

Let $M$ and $X$ be the spaces in the paper, let $p$ be the prime under consideration, and let $k$ be as above. Write $t_p^i(M)$ for the exponent of the $p$-primary torsion in $H^i(M,\operatorname{\mathbb{Z}})$, and let $\ell_+^i(X)$ and $\ell_-^i(X)$ denote the corresponding coefficients associated with $X$. Odd-degree torsion conjecture.

$$t_p^{2k+1}(M)=\sum_{j=0}^{k-1}\ell_+^{2j}(X)+\sum_{j=0}^{k-1}\ell_-^{2j+1}(X).$$

This conjecture proposes a precise way for the torsion to be distributed in $H^{2k+1}(M,\operatorname{\mathbb{Z}})$, complementing the preceding result for the sum of the torsion exponents in complementary odd degrees. Its status is not determined by the supplied source context.