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