🚀 go-pugleaf

Let P_n be the product of n copies of the 2-sphere S^2. In the
singular cohomology group H^2 (P_n, Z/rZ), let g_1, ..., g_n be the
generators corresponding to the factors of P_n. I have a topological
space E, an element x in H^2 (E, Z/rZ), and a continuous map f from
P_n to E such that f*(x) = g_1 + ... + g_n. Here E is an open subset
of a compact manifold, and even probably has finitely generated
cohomology. The object is to get lower bounds on the dimension of the
manifold that contains E, and which go to infinity when n goes to
infinity.

A while ago, I did something similar using Z coefficients instead of Z/rZ.
In this case, f*(x^n) = (g_1 + ... + g_n)^n, which is n! times a generator
of H^2n (P_n, Z) = Z. So H^2n (E, Z) is not zero. In the present case, r
is fixed (and presumably much smaller than n), but (g_1 + ... + g_n)^n
is zero as soon as r divides n!. If r is prime, this means the method only
works for n<r. In that case, if n=r then a cohomology operation called a
generalized Pontrjagin power, when applied to g_1 + ... + g_r, yields a
nonzero element of H^2r (P_r, Z/r^2Z). Therefore H^2r (E, Z/r^2Z) is not
zero. But even this seems not to be helpful for n>r.

What I am really hoping to find is a cohomology operation which one can
apply to g_1 + ... + g_n and get a nonzero element of H^2n (P_n, A) for
some coefficient group A, for n >> r. Of course, it doesn't have to be
exactly H^2n; anything close would help. Any other ideas would also be
appreciated. Of course, what I want may be false. If somebody can produce
for me an open subset U of R^5, say, and a continuous map f from
S^2 x S^2 x S^2 to U such that the restrictions of f to the sets
S^2 x point x point, point x S^2 x point, and point x point x S^2 are
all homotopically nontrivial, then I would probably give up on this
approach entirely.

Two additional items that might possibly be relevant: E is probably
simply connected, and H_2 (E, Z) is probably isomorphic to Z/rZ. (So
the U above should really be simply connected.)

Please reply to phillips@math.uoregon.edu. Thanks.