Rational toral rank of a map

Publication date: 9 November 2015

Abstract: Let

X

and

Y

be simply connected CW complexes with finite rational cohomologies. The rational toral rank

r_{0} (X)

of a space

X

is the largest integer

r

such that the torus

T^{r}

can act continuously on a CW-complex in the rational homotopy type of

X

with all its isotropy subgroups finite cite{H}. As a rational homotopical condition to be a toral map preserving almost free toral actions for a map

f : X o Y

, we define the rational toral rank

r_{0} (f)

of

f

, which is a natural invariant with

r_{0} (i d_{X}) = r_{0} (X)

for the identity map

i d_{X}

of

X

. We will see some properties of it by Sullivan models, which is a free commutative differential graded algebra over

Q

cite{FHT}.