- Stunted projective space
In
mathematics , a stunted projective space is a construction on aprojective space of importance inhomotopy theory . Part of a conventional projective space is collapsed down to a point.More concretely, in a
real projective space ,complex projective space orquaternionic projective space :"KP""n",
where "K" stands for the
real number s,complex number s orquaternion s, one can find (in many ways) copies of:"KP""m",
where "m" < "n". The corresponding stunted projective space is then
:"KP""n,m" = "KP""n"/"KP""m",
where the notation implies that the "KP""m" has been identified to a point. This makes a
topological space that is no longer amanifold . The importance of this construction was realised when it was shown that real stunted projective spaces arose asSpanier-Whitehead dual s of spaces ofIoan James , so-called "quasi-projective spaces", constructed fromStiefel manifold s. Their properties were therefore linked to the construction offrame field s onsphere s.In this way the
vector fields on spheres question was reduced to a question on stunted projective spaces: for R"P""n,m", is there a degree one mapping on the 'next cell up' (of the first dimension not collapsed ih the 'stunting') that extends to the whole space?Frank Adams showed that this could not happen, completing the proof.In later developments spaces "KP"∞,"m" and stunted
lens space s have also been used.
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