The algebraic toral rank conjecture

Let $(\Lambda V,{\operatorname{d}})$ be a minimal Sullivan algebra. Let

$$(\Lambda\langle t_1,\ldots,t_r\rangle,0)\hookrightarrow(\Lambda\langle t_1,\ldots,t_r\rangle\otimes\Lambda V,{\operatorname{d}})\to(\Lambda V,\bar{\operatorname{d}})$$

be a relative minimal model with $\deg t_i=2$ for $1\leq i\leq r$, and suppose that

$$H(\Lambda\langle t_1,\ldots,t_r\rangle\otimes\Lambda V,{\operatorname{d}})$$

is finite dimensional. Algebraic toral rank conjecture. Then

$$\dim H(\Lambda V,{\operatorname{d}})\geq 2^r.$$

This is the algebraic transcription of the toral rank conjecture via models of Borel fibrations. The paper explains its relation to almost free torus actions and gives counterexamples to the corresponding generalised toral-rank principle, so this assertion is not valid in the stated generality.