2 Construction of self maps
We wish to construct a map inducing multiplication by on homology. The strategy is to construct a map that induces multiplication by on homology, and then extend it to a map by obstruction theory.
First consider the Adams–Novikov spectral sequence for . In degrees up to , the spectral sequence looks like
where and . If , then we have an extra which will be right above .
In either case, we see that there is no room for extra differentials. So we see that
There is a map that induces multiplication by on . If , then this map has order and is unique.
If , then there is a map that induces multiplication by on .
In the case , we know or . If it is , then this map has order and does not lift to a map . This is indeed the case, as one can check using the Adams spectral sequence, so we do not have a self map at .