fiber sequence of the inclusion of CPⁿ into CP infinity

Let be the fiber of the inclusion.
Note that the skeleton inclusion is connected.
Furthermore is simply connected.
Furthermore of has homology in degrees and above.

So relative Hurewicz theorem tells us that the homotopy of the fiber

and it has to be zero before that.

Now consider the cohomological serre spectral sequence, here exemplarily for

Now the element entry must be killed, since .
Hence the map must be an isomorphism.

furthermore multiplication of a multiplicative spectral sequence under different pages tells us that must be nonzero in the page, since we will show that its differential under the leibniz rule will be nonzero:

The leibniz rule says up to sign

so in particular and each from the -th row must be an isomorphism - hence up until the

Furthemore can't have homology in degrees above since then can't be killed.

therefore nilpotent cohomology sphere is equivalent to a sphere shows that as desired

Consider the fiber sequence

It is a principal bundle (todo ) so by the classification theorem of principal g-bundles we get a pullback

for

But is contractible, so in particular the infinite sphere is contractible space,