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For $n \in \mathbb{N}$ with $n \gt 1$, consider continuous functions between spheres of the form
The homotopy cofiber of $f$ (the attaching space induced by $f$)
Hence for $\alpha_n, \beta_{2n}$ generators of the cohomology groups in degree $n$ and $2n$ (unique up to choice of sign), respectively, there exists an integer $HI(f)$ which expresses the cup product square of $\alpha_n$ as a multiple of $\beta_{2n}$:
This integer $HI(f) \in \mathbb{Z}$ is called the Hopf invariant of $f$ (e.g. Mosher-Tangora 86, p. 33).
This depends on the choices made only up to sign. In particular it has a well-defined mod-2 reduction image $[HI(f)] \in \mathbb{F}_2 = \mathbb{Z}/2\mathbb{Z}$ (in Z/2), and as such it is the Steenrod square
Here is a more abstract picture of the Hopf invariant in abstract homotopy theory (following SS21):
Let $E$ be a multiplicative cohomology theory, assumed to vanish in degree $2n - 1$
We write $E_n \;\coloneqq\; \Omega^\infty \Sigma^n E$ for its classifying spaces, hence for the component spaces of its representing spectrum.
(For the case of ordinary cohomology $E = H \mathbb{Z}$ we have $E_n \,\simeq\, K(\mathbb{Z},n)$ an Eilenberg-MacLane space.)
Now let
be a map (1). Then its $E$-Hopf invariant “is” the following homotopy pasting diagram of pointed homotopy types (see also at e-invariant is Todd class of cobounding (U,fr)-manifold):
the top square is defined to be a homotopy pushout, exhibiting the attaching space $C_f$;
the total rectangle is defined to be a homotopy pushout, exhibiting the suspension of $S^{2n-1}$ to $S^{2n}$;
the bottom square is hence a homotopy pushout by the pasting law;
by assumption (3) the restriction of $\Sigma^{2n} 1$ to $S^{2n-1}$ trivializes, exhibited by a choice of homotopy filling the full top part of the diagram; and by the universal property of the top homotopy-pushout this corresponds equivalently to the dashed morphism $c$;
the restriction of the cup square cohomology operation on $c$ to $S^n$ trivializes: $c$ factors through the $n$ component space of the connective cover $E\langle 0\rangle$, whence its cup square (by the discussion at connective cover – For ring spectra) factors through the $2n$-connective $2n$th component space of the same connective cover (as in a standard argument in complex oriented cohomology theory, e.g. Lurie, Lec. 4. Exmpl. 8, or see Conner-Floyd 66, Part I, Cor. 7.2):
This yields the homotopy filling the full bottom part of the diagram above; and by the universal property of the bottom homotopy-pushout this corresponds equivalently to a dashed morphism $S^{2n} \to E_{2n}$, labeled by some class
finally, by the homotopy-pushout property of the total rectangle, this class $\kappa$ also labels the total homotopy filling the full diagram.
We see that:
for $E = H \mathbb{Z}$ being ordinary cohomology, $\kappa \,\in\, \mathbb{Z}$ is the traditional Hopf invariant as discussed above;
for $E = KU$ being complex topological K-theory, $\kappa \,\in\, \mathbb{Z}$ is the Hopf invariant in K-theory as used in Adams-Atiyah 66.
In the case that the map $f$ is one the classical Hopf fibrations, the attaching space above is a projective space (by the discussion at cell structure of projective spaces) and the choice of homotopy $c$ is the choice of an orientation in $E$-cohomology theory to second stage. Specifically:
for the complex Hopf fibration $f = h_{\mathbb{C}}$ the attaching space is complex projective space $\mathbb{C}P^2$ and the choice of homotopy $c$ is a choice of complex orientation to second stage;
for the quaternionic Hopf fibration $f = h_{\mathbb{H}}$ the attaching space is quaternionic projective space $\mathbb{H}P^2$ and the choice of homotopy $c$ is a choice of quaternionic orientation to second stage.
Moreover, the de-composition of this pasting diagram exhibits the homotopy Whitehead integral/functional cup product-formula for the Hopf invariant:
For $n$ odd, the Hopf invariant necessarily vanishes. For $n$ even however, then there is a homomorphism
whose image contains at least the even integers.
A famous open question in the 1950s was which maps $f$ (1) have Hopf invariant one, namely $[HI(f)] = 1$ (2).
The Hopf invariant one theorem (Adams60) states that the only maps of Hopf invariant one, $[HI(f)] = 1$, are the Hopf constructions on the four real normed division algebras:
the real Hopf fibration;
By standard results in rational homotopy theory, every continuous function
corresponds to a unique dgc-algebra homomorphism
between Sullivan models of n-spheres.
The unique free coefficient of this homomorphism $CE(\mathfrak{l}f)$ is the Hopf invariant $HI(f)$ of $f$:
See at Whitehead integral formula and see the references below
Dale Husemöller, chapter 15 of: Fibre Bundles, Graduate Texts in Mathematics 20, Springer New York (1966) (doi:10.1007/978-1-4757-2261-1)
Robert Mosher, Martin Tangora, p. 33 of Cohomology operations and applications in homotopy theory, Harper & Row 1986 (pdf)
Michael Crabb, Andrew Ranicki, The Geometric Hopf Invariant and Surgery Theory, Springer Monographs in Mathematics, Springer (2017) $[$doi:10.1007/978-3-319-71306-9, slides pdf, pdf$]$
(in relation to surgery theory)
Klaus Wirthmüller, section 12 of: Vector bundles and K-theory, 2012 (pdf)
Gereon Quick, The Hopf invariant one problem via K-theory, lecture notes in: Advanced algebraic topology, 2014 (pdf)
See also:
And see the references at Hopf invariant one for more, in particular for the formulation via topological K-theory.
Discussion via differential forms/real homotopy theory/rational homotopy theory (see also at functional cup product):
J. H. C. Whitehead, An expression of Hopf’s invariant as an integral, Proc. Nat. Acad. Sci. USA 33 (1947), 117–123 (jstor:87688)
Hassler Whitney, Section 31 in Geometric Integration Theory, 1957 (pup:3151)
André Haefliger, p. 3 of Whitehead products and differential forms, In: P.A. Schweitzer (ed.), Differential Topology, Foliations and Gelfand-Fuks Cohomology, Lecture Notes in Mathematics, vol 652. Springer 1978 (doi:10.1007/BFb0063500)
Raoul Bott, Loring Tu, Prop. 17.22 in Differential Forms in Algebraic Topology, Graduate Texts in Mathematics 82, Springer 1982 (doi:10.1007/BFb0063500)
Lee Rudolph, Whitehead’s Integral Formula, Isolated Critical Points, and the Enhancement of the Milnor Number, Pure and Applied Mathematics Quarterly Volume 6, Number 2, 2010 (arXiv:0912.4974)
Phillip Griffiths, John Morgan, Section 14.5 of: Rational Homotopy Theory and Differential Forms, Progress in Mathematics Volume 16, Birkhauser (1981, 2013) (doi:10.1007/978-1-4614-8468-4)
Dev Sinha, Ben Walter, Lie coalgebras and rational homotopy theory II: Hopf invariants, Trans. Amer. Math. Soc. 365 (2013), 861-883 (arXiv:0809.5084, doi:10.1090/S0002-9947-2012-05654-6)
Felix Wierstra, Hopf Invariants in Real and Rational Homotopy Theory (2017) [diva:146246, pdf]
Domenico Fiorenza, Hisham Sati, Urs Schreiber, Twisted Cohomotopy implies M5 WZ term level quantization, Comm. Math. Phys. 2020 (arXiv:1906.07417)
Last revised on April 3, 2023 at 11:11:29. See the history of this page for a list of all contributions to it.