nLab f-invariant

Contents

Idea

The $f$ -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 $E$ .

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 $E$ -Adams spectral sequence.

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

(…)

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

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

(…)

The concept is due to

Gerd Laures, The topological $q$ -expansion principle, Topology Volume 38, Issue 2, March 1999, Pages 387-425 (doi:10.1016/S0040-9383(98)00019-6)
Gerd Laures, On cobordism of manifolds with corners, Trans. Amer. Math. Soc. 352 (2000) (doi:10.1090/S0002-9947-00-02676-3)

(see Genauer 12 for more on the bordism classes of manifolds with corners)

Further discussion in:

Hanno von Bodecker, On the geometry of the f-invariant (arXiv:0808.0428)
Mark Behrens, Gerd Laures, $\beta$ -family congruences and the f-invariant (arXiv:0809.1125)
Ulrich Bunke, Niko Naumann, The f-invariant and index theory, Manuscripta Math. 132, 365–397 (2010) (arXiv:0808.0257, https://doi.org/10.1007/s00229-010-0351-7)
Gerd Laures, Toda brackets and congruences of modular forms (arXiv:1102.3783, euclid:agt/1513715257)