On the degrees of equivariant maps from spheres to complex Stiefel manifolds

Abstract: We study the set of degrees of $\mathbb{Z}/m$-equivariant maps from spheres to complex Stiefel manifolds, motivated by the work of Astey--Gitler--Micha--Pastor. Under a suitable arithmetic condition, this set is determined using results of James, Atiyah--Todd, and Adams--Walker. Our approach is homotopy-theoretic.

• The paper proves that, when the Hurewicz index h_{n,k}(r)=1, the equivariant degree set is either empty or the full residue class a≡binom(n,r) mod m.
• The authors compute the relevant projective Stiefel cohomology, derive the necessary binomial congruence, and show obstruction-based degree modification produces infinitely many realizations.
• The paper establishes existence for r=n−k+1 for every m and for even r=n−k+2 under stated binomial divisibility conditions, while leaving broader existence questions open.

This paper, by Haibao Duan and Ruizhi Huang, determines the set of degrees realized by $\mathbb{Z}/m$-equivariant maps from odd-dimensional spheres into complex Stiefel manifolds, under an arithmetic condition on the parameters. The work is motivated by the program of Astey–Gitler–Micha–Pastor on triviality indices of direct sums of the canonical line bundle over lens spaces, and its method is entirely homotopy-theoretic.

Background and the degree sets

For the complex Stiefel manifold $V_{n,k}(\mathbb{C})$ of orthonormal $k$-frames in $\mathbb{C}^n$, Borel's computation gives an integral cohomology ring that is an exterior algebra $\Lambda(x_{2(n-k+1)-1},\ldots,x_{2n-1})$. For $r$ in the index set $I_{n,k}=\{n-k+1,\ldots,n\}$ and a map $f\colon S^{2r-1}\to V_{n,k}(\mathbb{C})$, the integer defined by $f^\ast(x_{2r-1})=\deg(f)\cdot\omega_{S^{2r-1}}$ is called the degree of $f$, and the set of all such degrees is denoted $V_{n,k}(\mathbb{C})$0. Equipping both sphere and Stiefel manifold with their standard fixed-point-free $V_{n,k}(\mathbb{C})$1-actions yields the equivariant analogue

$V_{n,k}(\mathbb{C})$2

The central object controlling these sets is the Hurewicz index $V_{n,k}(\mathbb{C})$3: since each exterior generator $V_{n,k}(\mathbb{C})$4 is rationally spherical (the minimal Sullivan model has zero differential), $V_{n,k}(\mathbb{C})$5 contains a $V_{n,k}(\mathbb{C})$6-summand, and $V_{n,k}(\mathbb{C})$7 is the absolute value of the Hurewicz image of a generator. Naturality of the Hurewicz homomorphism shows immediately that every degree is a multiple of $V_{n,k}(\mathbb{C})$8, and a map of degree exactly $V_{n,k}(\mathbb{C})$9 exists; hence

$k$0

A reduction lemma via the fibration $k$1 shows that $k$2, so all computations reduce to the top-dimensional case.

Characterization of the unit index via Adams–Walker

The key arithmetic input is the classical lifting problem: $k$3 precisely when there is a lift in a diagram involving the Stiefel fibration $k$4 and a degree-one self-map of the sphere — equivalently, when $k$5 admits a cross-section. By the theorem of Adams–Walker, this holds if and only if the Atiyah–Todd number $k$6 divides $k$7, where $k$8 for primes $k$9. This yields concrete sufficient conditions: $\mathbb{C}^n$0 whenever $\mathbb{C}^n$1, whenever $\mathbb{C}^n$2 is even, or whenever $\mathbb{C}^n$3 with $\mathbb{C}^n$4.

The integral cohomology of $\mathbb{C}^n$5

The equivariant argument passes through the complex projective Stiefel manifold $\mathbb{C}^n$6, quotient of $\mathbb{C}^n$7 by the diagonal free $\mathbb{C}^n$8-action. The authors establish the integral cohomology ring as an epimorphic image of $\mathbb{C}^n$9, where

$\Lambda(x_{2(n-k+1)-1},\ldots,x_{2n-1})$0

and $\Lambda(x_{2(n-k+1)-1},\ldots,x_{2n-1})$1 for $\Lambda(x_{2(n-k+1)-1},\ldots,x_{2n-1})$2. Notably, the paper remarks that an earlier statement of this ring due to Ruiz is incorrect in the form needed here. The proof proceeds by Serre spectral sequence analysis of the principal circle bundle, using Borel's transgression formula $\Lambda(x_{2(n-k+1)-1},\ldots,x_{2n-1})$3 and ruling out additional torsion generators via the mod-$\Lambda(x_{2(n-k+1)-1},\ldots,x_{2n-1})$4 computations of Astey–Gitler–Micha–Pastor (and, for $\Lambda(x_{2(n-k+1)-1},\ldots,x_{2n-1})$5, Duan's computation for projective unitary groups).

Necessary congruence and sufficiency by obstruction modification

The pivotal necessary condition is obtained from the Gysin sequence of the relevant circle bundles: any $\Lambda(x_{2(n-k+1)-1},\ldots,x_{2n-1})$6-equivariant map $\Lambda(x_{2(n-k+1)-1},\ldots,x_{2n-1})$7 factors through an $\Lambda(x_{2(n-k+1)-1},\ldots,x_{2n-1})$8-equivariant map $\Lambda(x_{2(n-k+1)-1},\ldots,x_{2n-1})$9, and comparing Gysin sequences forces

$r$0

Two further consequences of independent interest are recorded: $r$1 relating the induced map on $r$2, and $r$3.

For sufficiency, the authors study a lifting problem over $r$4: a solution $r$5 induces a $r$6-equivariant map via $r$7. Using the coaction of $r$8 on itself and the fact that $r$9 acts through multiplication by $I_{n,k}=\{n-k+1,\ldots,n\}$0 on the $I_{n,k}=\{n-k+1,\ldots,n\}$1-summand of $I_{n,k}=\{n-k+1,\ldots,n\}$2, they show that degrees can be modified by multiples of $I_{n,k}=\{n-k+1,\ldots,n\}$3. Consequently:

• If $I_{n,k}=\{n-k+1,\ldots,n\}$4 and some equivariant map exists, then $I_{n,k}=\{n-k+1,\ldots,n\}$5.
• In general, the equivariant degree set, when nonempty, is always infinite — it can never be a finite nonempty set.

The main theorem combines these observations: the equivariant degree set is either empty or a full residue class modulo $I_{n,k}=\{n-k+1,\ldots,n\}$6 centered at $I_{n,k}=\{n-k+1,\ldots,n\}$7, whenever $I_{n,k}=\{n-k+1,\ldots,n\}$8.

Two existence results

Case $I_{n,k}=\{n-k+1,\ldots,n\}$9. A geometric construction via a lifting over $f\colon S^{2r-1}\to V_{n,k}(\mathbb{C})$0 produces an $f\colon S^{2r-1}\to V_{n,k}(\mathbb{C})$1-equivariant map $f\colon S^{2r-1}\to V_{n,k}(\mathbb{C})$2 of degree $f\colon S^{2r-1}\to V_{n,k}(\mathbb{C})$3, verified by transgression naturality in a morphism of Serre spectral sequences. Hence the equivariant degree set equals $f\colon S^{2r-1}\to V_{n,k}(\mathbb{C})$4 for every $f\colon S^{2r-1}\to V_{n,k}(\mathbb{C})$5 — no restriction on $f\colon S^{2r-1}\to V_{n,k}(\mathbb{C})$6 beyond the congruence itself.

Case $f\colon S^{2r-1}\to V_{n,k}(\mathbb{C})$7 even. Under the divisibility hypotheses $f\colon S^{2r-1}\to V_{n,k}(\mathbb{C})$8 (equivalently $f\colon S^{2r-1}\to V_{n,k}(\mathbb{C})$9 on $f^\ast(x_{2r-1})=\deg(f)\cdot\omega_{S^{2r-1}}$0), successive lifts through $f^\ast(x_{2r-1})=\deg(f)\cdot\omega_{S^{2r-1}}$1 and $f^\ast(x_{2r-1})=\deg(f)\cdot\omega_{S^{2r-1}}$2, using Corollary 5.12 of Astey–Gitler–Micha–Pastor, yield a $f^\ast(x_{2r-1})=\deg(f)\cdot\omega_{S^{2r-1}}$3-equivariant map, and again the full residue class is realized. This case relies on the equivalence between equivariant maps and triviality of a rank-$f^\ast(x_{2r-1})=\deg(f)\cdot\omega_{S^{2r-1}}$4 subbundle of $f^\ast(x_{2r-1})=\deg(f)\cdot\omega_{S^{2r-1}}$5 over $f^\ast(x_{2r-1})=\deg(f)\cdot\omega_{S^{2r-1}}$6, established via the pullback diagram to $f^\ast(x_{2r-1})=\deg(f)\cdot\omega_{S^{2r-1}}$7 and $f^\ast(x_{2r-1})=\deg(f)\cdot\omega_{S^{2r-1}}$8.

By contrast, the paper recalls that for $f^\ast(x_{2r-1})=\deg(f)\cdot\omega_{S^{2r-1}}$9 with $f$0 and $f$1 odd if $f$2 is even, no equivariant map exists — so emptiness genuinely occurs outside the covered cases.

Limitations and open questions

The general characterization of $f$3 remains incomplete: Theorem 1.1 determines the equivariant degree set only when $f$4, i.e., when $f$5, and even then requires an existence input. Existence is settled here only for $f$6 and for $f$7 even under binomial-coefficient divisibility assumptions; for other values of $f$8, deciding emptiness versus realization of the residue class is open. The paper also leaves a structural question about the decompositions $f$9 produced by Proposition: although infinitely many such decompositions arise from equivariant maps of pairwise distinct degrees, the homotopy classes of their classifying maps lie in the finite set $V_{n,k}(\mathbb{C})$00 (since $V_{n,k}(\mathbb{C})$01 is torsion). It is unknown which pairs $V_{n,k}(\mathbb{C})$02 give isomorphic bundles $V_{n,k}(\mathbb{C})$03, and whether there exist rank-$V_{n,k}(\mathbb{C})$04 bundles over $V_{n,k}(\mathbb{C})$05 not isomorphic to any $V_{n,k}(\mathbb{C})$06.

Conclusion

The paper reduces the classification of equivariant mapping degrees from spheres into complex Stiefel manifolds to two tractable problems: the arithmetic of the Atiyah–Todd numbers, resolved by Adams–Walker, and the existence of equivariant maps, settled here in two special cases. The resulting picture — equivariant degree sets are either empty or entire residue classes modulo $V_{n,k}(\mathbb{C})$07 — sharpens the earlier work of Astey–Gitler–Micha–Pastor and supplies new tools, notably the corrected integral cohomology of $V_{n,k}(\mathbb{C})$08 and the degree-modification technique, applicable to further instances of the problem.