nonzero homological degree and surjectivity on the disk

Let be a continuous map on the disk, such that

and the degree of the restriction is nonzero

Then is surjective

Suppose is not surjective, then there exists a point such that . Then define a map

it follows, that

and therefore has nonzeroo degree.

Furthermore, it factorizes as

applying the singular homology functor gives

which is a contradiction, as , but it doesnt factor