I'm a grad student studying non-negative curvature on simply-connected manifolds and the conjectured relationship with rational homotopy theory.
The rational homotopy groups (and so the number of generators in the Sullivan minimal model) of the connected sum $\mathbb{CP}^2 \# \mathbb{CP}^2 \# \mathbb{CP}^2$ (both positive orientation gluings) grow exponentially as a function of degree. I am interested in calculating the actual exponent of the growth, or understanding it asymptotically, or at least getting the largest lower bound for it that I can.
Here is my current strategy: because minimal models are constructed inductively in "phases", I thought I could find and prove an inductive pattern in the way new generators are added to destroy cohomology that occurs past degree 4. A good reference for the inductive construction of Sullivan minimal models is Felix-Halperin-Thomas, Rational Homotopy Theory 2001, pp. 141. In their notation, what I am calling the "phase" is $k \in \mathbb{N}$ where the Sullivan condition is $d:V_k \rightarrow \Lambda (\bigoplus_{i=0}^{k-1} V_i)$.
I am working with the Sage CDGA package to experiment with patterns. To simplify the problem, I've made a CDGA (phase 0) with 3 cocycles in degree 2, then 5 generators in degree 3 whose differentials impose the relations necessary to get the right cohomology up through degree 4. This CDGA has lots of cohomology past degree 4, so is not a model for my manifold.
Then, in phase 1 of the construction, I made one additional generator $z$ in degree 4 whose differential destroys the first degree 5 co-cycle that shows up from phase 0. This is a smallest first step towards a minimal model.
I wanted to propagate through the consequences of this new generator hoping that I could show that it creates new co-cycles that are necessarily not co-boundaries because they have a $z$ in their expression, and the Sullivan condition on the differential says that the image of the differential of phase 1 generators must land in phase 0.
I thought maybe there would be some way to take a quotient that isolated these $z$-containing co-cycles I'm looking for.
I'm struggling to get this to work, and am wondering if folks have any other leads I could look into for understanding the exponential growth quantitatively, or ideas for how I could get my idea to work. Thanks.
