There are manifolds (rationally elliptic) $M_1$ and $M_2$ of different rational homotopy types but their rational cohomology algebras coincide. Such examples were discussed in Nishimoto, Shiga, Yamaguchi's joint work ("Elliptic rational space whose cohomologies and homotopies are isomorphic"). In this paper, they defined a family of minimal algebras and proved that their cohomology algebras are the same but rational homotopy types are different.

However, the minimal algebras they have considered have generators of the same degree. i.e. both the minimal algebras have the same generating sets but the differentials are different.

Is it possible to have two minimal algebras (rationally elliptic) having different degree generators but their cohomology algebras agree?

  • 1
    $\begingroup$ To answer the problem in the title, you can take minimal models of the smallest simply connected non-formal manifold, i.e. the sphere bundle of a real rank 4 real bundle over $S^2 \times S^2$ with nontrivial Euler class, with model $(\Lambda(x_2, x'_2, y_3, y'_3, z_3), dy = x^2, dy' = x'^2, dz = xy)$, and of $(S^2 \times S^5)\# (S^2 \times S^5)$ (which is the formal manifold with the same cohomology). The minimal model of the latter has generators of degree 4, 5, 6, ... . It is also not elliptic, and I don't know the answer now if you want both spaces elliptic. $\endgroup$ Mar 28 at 13:32
  • 1
    $\begingroup$ Sorry, the differential of $z$ in the first model should be $xx’$, not $xy$ $\endgroup$ Mar 29 at 20:26
  • $\begingroup$ Thank you so much!! $\endgroup$
    – piper1967
    Mar 29 at 21:45
  • 1
    $\begingroup$ The generators of the minimal model of the simply connected space $X$ are dual to the cohomology of the Harrison complex $Harr(A^*(X))$. If you're interested in elliptic spaces, both cohomology and homotopy groups are finite dimensional, so there are no double-dualization related problems arise. As such, if you have two spaces, one $X_1$ formal and another one $X_2$ not, you will have a spectral sequence with non-trivial higher differentials converging from $\pi_*(X_1)$ to $\pi_*(X_2)$. $\endgroup$ Apr 1 at 22:22
  • 1
    $\begingroup$ So any formal elliptic space, which is not intrinsically formal (so there is another, formal space with the same cohomology but different rational homotopy type), should give you an example of such a couple. I can't think of an example right out of my head, but there should be some in the literature $\endgroup$ Apr 1 at 22:31


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy