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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?

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up vote 1 down vote accepted

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.

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