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It seems that both $RP^2$ and $RP^3$ have the same fundamental group $Z_2$, but Why no map from $RP^3 \to RP^2$ induces an isomorphism between their fundamental groups?

John0417
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1 Answers1

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This follows more or less directly by the Borsuk-Ulam theorem.

Suppose such a map existed, the by covering theory you know that a map which induces an isomorphism on $\pi_1$ lifts to an equivariant map between the universal coverings.

Here this simply means there would exist a $\mathbb{Z}_2$ equivariant map $S^3 \to S^2$ where the action is given by multiplying with $1$ or $-1$.

But the Borduk Ulam theorem says that such an equivariant map does not exist, hence there is no map between these projective spaces inducing an isomorphism on the fundamental group.

mland
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