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f-invariant

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Contents

Idea

The ff-invariant (Laures 99) is the third in a sequence of homotopy invariants of “stable maps”, i.e. of morphisms in the stable homotopy category (in particular: of stable homotopy groups of spheres), being elements of Ext-groups between the homology groups/cohomology groups of the domain and codomain of the map, with respect to some suitable choice of Whitehead-generalized cohomology theory EE.

The previous two invariants in the sequence are the d-invariant and the e-invariant. All these are elements that appear on the second page of the EE-Adams spectral sequence.

In higher analogy to how the e-invariant exists if the d-invariant vanishes and then makes sense for E=KUE = KU (complex topological K-theory), so the f-invariant exists when the e-invariant vanishes and then makes sense of EE an elliptic cohomology theory.

Moreover, in higher analogy to how the e-invariant, when expressed in terms of MU-bordism theory, computes the Todd genus/A-hat genus of a manifold with corners (see there), so the f-invariant computes an elliptic genus of a manifold with corners (Laures 00).

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Let XϕYX \overset{\phi}{\longrightarrow} Y be morphism in the stable homotopy category out of a finite spectrum XX (for instance the image under suspension Σ \Sigma^\infty of a morphism in the classical homotopy category of pointed homotopy types out of a finite CW-complex).

If EE be a multiplicative cohomology theory satisfying the flatness assumptions used in the Adams spectral sequence, and such that the e-invariant of ϕ\phi in EE vanishes. Then the ff-invariant of ϕ\phi (Laures 99) is a certain element in the second Ext-group Ext E (E) 2(E (X),E (Y))Ext^2_{E_\bullet(E)}\big( E_\bullet(X), E_\bullet(Y) \big).

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References

The concept is due to

Further discussion in:

See also:

Last revised on March 8, 2021 at 02:55:20. See the history of this page for a list of all contributions to it.