Given a closed
n
n
-manifold
M
n
{M^n}
and a tuple of positive integers
P
P
, let
σ
P
M
{\sigma _P}M
be the
P
P
-spin of
M
M
. If
M
n
⋍̸
S
n
{M^n} \not \backsimeq {S^n}
and
P
≠
Q
P \ne Q
(as unordered tuples), it is shown that
σ
P
M
⋍̸
σ
Q
M
{\sigma _P}M\not \backsimeq {\sigma _Q}M
if either (1)
H
∗
(
M
n
)
≇
H
∗
(
S
n
)
{H_*}({M^n})\not \cong {H_*}({S^n})
, (2)
π
1
M
{\pi _1}M
finite, (3)
M
M
aspherical, or (4)
n
=
3
n = 3
. Applications to the homotopy classification of homology spheres and knot exteriors are given.