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Kervaire invariants and selfcoincidences
, 2007
"... Abstract. Minimum numbers decide whether a given map f : S m → S n /G from a sphere into a spherical space form can be deformed to a map f such that f (x) = f (x) for all x ∈ S m . In this paper we compare minimum numbers to (geometrically defined) Nielsen numbers (which are more computable). In th ..."
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Abstract. Minimum numbers decide whether a given map f : S m → S n /G from a sphere into a spherical space form can be deformed to a map f such that f (x) = f (x) for all x ∈ S m . In this paper we compare minimum numbers to (geometrically defined) Nielsen numbers (which are more computable). In the stable dimension range these numbers coincide. But already in the first nonstable range (when m = 2n − 2) the Kervaire invariant appears as a decisive additional obstruction which detects interesting geometric coincidence phenomena. Similar results (involving e.g. Hopf invariants, taken mod 4) are obtained in the next seven dimension ranges (when 1 < m − 2n + 3 8). The selfcoincidence context yields also a precise geometric criterion for the open question whether the Kervaire invariant vanishes on the 126stem or not.
The geometric Hopf invariant and double points
, 2010
"... The geometric Hopf invariant of a stable map F is a stable Z/2equivariant map h(F) such that the stable Z/2equivariant homotopy class of h(F) is the primary obstruction to F being homotopic to an unstable map. In this paper we express the geometric Hopf invariant of the Umkehrmap F of an immersi ..."
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The geometric Hopf invariant of a stable map F is a stable Z/2equivariant map h(F) such that the stable Z/2equivariant homotopy class of h(F) is the primary obstruction to F being homotopic to an unstable map. In this paper we express the geometric Hopf invariant of the Umkehrmap F of an immersion f: M m � N n in terms of the double point set of f. We interpret the SmaleHirschHaefliger regular homotopy classification of immersions f in the metastable dimension range 3m < 2n−1 (when a generic f has no triple points) in terms of the geometric Hopf invariant.