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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 $n=3,7,....,4m-1$.)

user168639
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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)

Dan Rust
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  • I an not able to verify that the product of n-sphere and double comb space is not homotopic to an n-sphere. – user168639 Sep 03 '14 at 14:33
  • I can't immediately see a rigorous argument, because all homotopy and (co)homology groups will be the same (by construction). It should be fairly easy to at least see that the space doesn't deformation retract onto one of the underlying spherical projections, using an argument which is similar to why the double comb space itself isn't contractible (See this question for proofs of this fact). – Dan Rust Sep 03 '14 at 15:40
  • As said in my answer, it might be easier to just consider the wedge product of the two spaces. – Dan Rust Sep 03 '14 at 15:45