example of weakly homotopic sphere

Question

There are spaces such as pseudocircles that are weakly homotopic to sphere but are not homotopic to spheres. But pseudo circles are non-hausdorff spaces. I need an example of a paracompact hausdorff space which is weakly homotopic to a n-sphere but not homotopic to a n-sphere. (preferably $n=3,7,....,4m-1$.)

Answer 1

Take the $n$-sphere and take its product with the double comb space which as far as I can see will satisfy all the necessary properties because the double comb space is a weakly contractible but not contractible space, and as a subset of the plane it is automatically Hausdorff and paracompact in lieu of Stone's theorem which says all metrisable spaces are paracompact.

It's possible a wedge product with the sphere will be a bit easier to work with (with the base point at one of the locally compact points of the combspace)