When one say two cohomology rings are isomorphic, does the degree of generator matters?

Question

My question is when one say two cohomology rings are isomorphic, does the degree of generator matters?

Here is a motivating example:

$H^*(RP^{\infty}, \mathbb Z_2)\cong \mathbb Z_2[\alpha]$ with $|\alpha|=1$, and $H^*(CP^{\infty}, \mathbb Z_2)\cong \mathbb Z_2[\beta]$, with $|\beta|=2$, but if we define map between two cohomology rings by sending $\alpha$ to $\beta$, it is clearly a ring isomorphism. But we know that the two spaces even don't share the same cohomology groups. So, I have a doubt on definition of isomorphism between cohomology rings.

Never mind, I have find this post useful: http://math.stackexchange.com/questions/1581681/why-does-dimension-of-the-generator-matter-for-cohomology-rings?rq=1 — AG learner, Mar 05 '16 at 23:42
What do you expect them to be isomorphic as? As rings sure theyre is isomorphic. As graded rings no of course not (a morphism of graded rings preserves the graded rings). A map between topological spaces induces a graded ring map between their cohomology rings. — PVAL-inactive, Mar 05 '16 at 23:42

score 2 · Accepted Answer · answered Mar 06 '16 at 00:27

2

Yes, you should ask for an isomorphism of graded rings.

answered Mar 06 '16 at 00:27

Qiaochu Yuan

419,620

When one say two cohomology rings are isomorphic, does the degree of generator matters?

1 Answers1