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Integral mapping-cone Hopf invariant for maps S3 to S2

Definition

Let $f : S^{3} \to S^{2}$ be a based continuous map. Give $S^{2}$ an orientation and give the attached $4$ -cell its chosen orientation. Its mapping cone is the CW complex $C_{f} = S^{2} \cup_{f} D^{4}$ , with the basepoint as its $0$ -cell. Let $x \in H^{2} (C_{f}; Z)$ be the unique class restricting to the positive generator of $H^{2} (S^{2}; Z)$ , and let $y \in H^{4} (C_{f}; Z)$ be the generator determined by the oriented $4$ -cell. Define the integral mapping-cone Hopf invariant $H (f) \in Z$ by the equation $x ⌣ x = H (f) y .$

Definition references

Hatcher, Algebraic Topology, Appendix §4.B

May, A Concise Course in Algebraic Topology, Chapter 7

Facts & Assumptions

Given: The based map, orientations, and coefficient group in the Definition. Use ZF + AC for the imported cohomology suppliers.

[F1]

The ordinary cell cone $C_{f} = S^{2} \cup_{f} D^{4}$ attaches the cone base by $f$ and collapses its tip, without collapsing the basepoint track. Mapping cylinder and mapping cone In the coned CW structure that track is a contractible subcomplex; collapsing it gives the reduced cone and is a based homotopy equivalence by CW quotients and collapse of a contractible subcomplex. The reduced convention is Reduced cone suspension and cofiber sequence. The ordinary attachment has the three-cell structure used below.

[F2]

Singular cohomology has a natural cup product $H^{p} (C_{f}; Z) \times H^{q} (C_{f}; Z) \to H^{p + q} (C_{f}; Z)$ . Singular cohomology ring

[F3]

Cellular cochains compute singular cohomology of a finite CW complex with the stated constant coefficients. Cellular cochains compute cohomology with local coefficients

[A1]

The $0$ -, $2$ -, and $4$ -cells of $C_{f}$ carry the orientations specified in the Definition.

Verification

1.1F1F3A1algebra

By [F1], the cells of $C_{f}$ have dimensions $0, 2, 4$ . Its cellular cochain groups are therefore $Z$ in those degrees and zero in degrees $1$ and $3$ ; every cellular coboundary into or out of degrees $2$ and $4$ is zero because the adjacent cochain group is zero. [F3] gives $H^{2} (C_{f}; Z) ≅ Z$ and $H^{4} (C_{f}; Z) ≅ Z$ , with the restriction and oriented top-cell generators in the Definition.

2.1F2step 1.1A1algebra∎

By [F2], $x ⌣ x$ belongs to $H^{4} (C_{f}; Z) = Z y$ . Since $y$ is a generator, there is a unique integer $H (f)$ with $x ⌣ x = H (f) y$ . The orientations fix its sign, so the displayed equation defines an integer without any further choice.

Integral mapping-cone Hopf invariant for maps S3 to S2

Definition

Definition references

Facts & Assumptions

Verification

Depends on

Dependency tree · 2 levels

Sources