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Nick L
Nick L
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With regards to the second question, if $Y$ is a $K(\pi,1)$ space then the natural map from $[X,Y]$ is isomorphic to $Hom(\pi_{1}(X),\pi_{1}(Y))$ divided (divided out by conjugacy) is a bijection (see the answers here Question about maps of $S^{3}$-bundles).

But there are examples where quite the opposite occurs, take $X = Y = S^{2}$, both are simply connected but there are infinitely many non-homotopic maps.

With regards to the second question, if $Y$ is a $K(\pi,1)$ space then $[X,Y]$ is isomorphic to $Hom(\pi_{1}(X),\pi_{1}(Y))$ divided out by conjugacy (see the answers here Question about maps of $S^{3}$-bundles).

But there are examples where quite the opposite occurs, take $X = Y = S^{2}$, both are simply connected but there are infinitely many non-homotopic maps.

With regards to the second question, if $Y$ is a $K(\pi,1)$ space then the natural map from $[X,Y]$ to $Hom(\pi_{1}(X),\pi_{1}(Y))$ (divided out by conjugacy) is a bijection (see the answers here Question about maps of $S^{3}$-bundles).

But there are examples where quite the opposite occurs, take $X = Y = S^{2}$, both are simply connected but there are infinitely many non-homotopic maps.

Source Link
Nick L
Nick L
  • 7k
  • 1
  • 15
  • 41

With regards to the second question, if $Y$ is a $K(\pi,1)$ space then $[X,Y]$ is isomorphic to $Hom(\pi_{1}(X),\pi_{1}(Y))$ divided out by conjugacy (see the answers here Question about maps of $S^{3}$-bundles).

But there are examples where quite the opposite occurs, take $X = Y = S^{2}$, both are simply connected but there are infinitely many non-homotopic maps.