Grothendieck tried to prove something (standard conj?) by “every variety is birational to…” (something ruled??), which I think is false. (Perhaps this related to products of curves, and/or devissage). Can we use a similar idea, but with -homotopy instead of birationality? How are teh two concepts related? Note: Birational implies same dimension.
Would it be possible to look for algebraic cycles using numerical/Monte Carlo/optimization methods? One could consider a space of polynomials, and try to vary the coeffs so as to find for example a curve lying on some given surface, satisfying some constraints.
Q: Is there for some (collection of) CTs an analogue of the theorem (Whitehead?) isomorphism on cohomology implies htpy equiv?
nLab page on Some ideas V