at.algebraic topology - Examples of the varying strengths of topological invariants

For 1 and 2, consider the $S^2$ bundles over $S^4$ (with structure group SO(3)). Using clutching functions, one can see that there is a $mathbb{Z}$s worth of such bundles indexed by, say, k.

Using the Gysin sequence, one finds that the cohomology groups (and homology groups) are the same as those of $S^2times S^4$, mainly, a $mathbb{Z}$ in dimension 0,2,4, and 6, and 0 elsewhere.

However, for $kneq pm k'$ the bundles corresponding to $k$ and $k'$ have nonisomorphic ring structures.

By Poincare duality, the only question about the ring structure is the following: what is the square of the degree 2 generator? Turns out, the square of the degree two generator is equal to $pm k$ times the square of the degree 4 generator. Incidentally, the case $k=1$, one gets the cohomology ring structure of $mathbb{C}P^3$. In fact, the total space of the bundle is diffeomorphic to $mathbb{C}P^3$.

For your third question, I'd inspect $S^5$ bundles over $S^2$. Again, by clutching function analysis, there must be precisely two such bundles - the trivial bundle and one other. By the Gysin sequence, these must have the same cohomology groups and by Poincare duality, the ring structures must in fact agree.

However, the second Stieffel Whitney class of the trivial bundle is trivial, while the second Stieffel Whitney class of the nontrivial bundle is nontrivial, and hence the two total spaces are not homotopy equivalent. (Stieffel Whitney classes are closely related to Steenrod operations).

And just to anticipate, the spaces $S^3times mathbb{R}P^2$ and $S^2times mathbb{R}P^3$ have all the same homotopy groups, but are not homotopy equivalent (as homology will tell you).

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