On spaces of maps between complex projective spaces

Abstract:

When $1 \leqslant m \leqslant n$, the space $M({P^m},{P^n})$ of maps of complex projective $m$-space ${P^m}$ into complex projective $n$-space ${P^n}$ has a countably infinite number of components enumerated by degrees of maps in ${H^2}({P^m};{\mathbf {Z}})$. By calculating their $(2n - 2m + 1)$-dimensional integral homology group we show that two components of $M({P^m},{P^n})$ are homotopy equivalent if and only if their associated degrees have the same absolute value.