High-corank torsion in homotopy of unitary groups via topological modular forms and higher real $K$-theories

Abstract: Work of Toda identifies groups of metastable vector bundles on even-dimensional spheres with stable homotopy groups of certain stunted projective spectra. Using Weiss' unitary calculus, the second-named author generalized this identification to show that metastable, stably trivial vector bundles on even cell complexes can naturally be identified with stable homotopy classes of maps into a shifted stunted projective spectrum. Thus, certain classical questions about vector bundles (or homotopy of unitary groups) can be rephrased as stable computations. In this note, we show that certain generalized cohomology theories arising in chromatic and equivariant homotopy theory can be used to deduce the existence of non-trivial, stably trivial vector bundles on spheres and complex projective spaces.

• The paper develops splitting lemmas that combine orientation orders with nonzero Hurewicz images in generalized cohomology theories to detect torsion without computing entire unstable homotopy groups.
• It produces explicit 2-, 3-, and p-torsion families in homotopy groups of unitary groups, including infinite families at primes 2 and 3 and finite families for each prime p≥5.
• The method extends detection beyond previous projective-space corank bounds, yielding nontrivial stably trivial bundles while leaving divisibility, group structure, and several odd-prime cases unresolved.

The paper by Chatham, Hu, and Opie (2608.19411) uses "designer" generalized cohomology theories—real $K$-theory $KO$, $2$-local topological modular forms $tmf_{(2)}$, and the higher real $K$-theories $EO_{p-1}$ of height $p-1$—to detect new families of $p$-power torsion in unstable homotopy groups of unitary groups $\pi_{2n}BU(r)$, and, via pullback along the projection onto the top cell, nontrivial stably trivial vector bundles on complex projective spaces in coranks beyond previously treated ranges. The method deliberately trades complete computation for detection: rather than computing $\pi_*^s CP_r^n$ outright, the authors exhibit specific nonzero composites after base change to a ring spectrum $KO$0 whose Hurewicz image is understood.

Homotopy-theoretic identification of metastable bundles

The starting point is the classical Toda identification: for odd $KO$1 in the metastable range,

$KO$2

where $KO$3 is the cofiber of the inclusion $KO$4. The second author's unitary-calculus refinement [Hu] upgrades this: for any finite even cell complex $KO$5 with cells up to dimension $KO$6 that is cohomologically even (or satisfies $KO$7), stably trivial rank-$KO$8 bundles on $KO$9 are identified with stable maps $2$0. For $2$1 this recovers Toda; for $2$2 it gives

$2$3

valid in the metastable range $2$4.

James periodicity implies these groups depend only on $2$5 modulo the James number $2$6 once the corank $2$7 is fixed. For $2$8, full computations exist; for $2$9 the period exceeds 2880, making systematic computation unwieldy. This motivates the detection strategy adopted here.

The detection mechanism and orientation order

The core lemma is elementary but effective. If $tmf_{(2)}$0 is an $tmf_{(2)}$1-ring spectrum such that $tmf_{(2)}$2 (a multiple of the tautological bundle) is $tmf_{(2)}$3-orientable, and $tmf_{(2)}$4 has nonzero Hurewicz image in $tmf_{(2)}$5, then for suitable rank and dimension constraints the composite through the bottom cell of $tmf_{(2)}$6 defines a nontrivial vector bundle. The orientability hypothesis forces the relevant cell of $tmf_{(2)}$7 to split, so nonvanishing reduces to nonvanishing of the $tmf_{(2)}$8-Hurewicz image of $tmf_{(2)}$9. The required orientability facts are supplied by Bhattacharya–Chatham's theory of orientation order ($K$0, $K$1, $K$2 at height $K$3) and by Chatham's proof that $K$4 in the height-$K$5 case.

Results for spheres

The main theorem produces four infinite or parametrized families:

Source	Prime	Family
$K$6	2	Nonzero 2-torsion in $K$7 for all $K$8, $K$9
$EO_{p-1}$0	2	Nonzero 2-torsion in $EO_{p-1}$1, $EO_{p-1}$2 for $EO_{p-1}$3, $EO_{p-1}$4, using classes $EO_{p-1}$5 and $EO_{p-1}$6 from Behrens–Mahowald–Quigley
$EO_{p-1}$7	3	Nonzero 3-torsion in $EO_{p-1}$8 for $EO_{p-1}$9, using Belmont–Shimomura's classes $p-1$0
$p-1$1, $p-1$2	$p-1$3	Nonzero $p-1$4-torsion in $p-1$5 for $p-1$6, $p-1$7

For $p-1$8 the Hurewicz image in $p-1$9 is only known to contain the finite family $p$0 detected by $p$1 in the Novikov spectral sequence (Ravenel, Nave); the authors note the image is widely suspected to be finite in that range, so the resulting torsion families are finite per prime. At $p$2 the image is infinite, giving genuinely infinite families.

A warm-up subsection shows that when $p$3, the $p$4-homology computation combined with Hurewicz surjectivity detects $p$5-torsion already covered by Matsunaga's classical theorem—and does not recover its $p$6-divisibility information. The authors observe that identifying their classes as permanent cycles and inspecting Adams extensions could yield an alternate proof of Matsunaga's result, though they do not carry this out.

Results for projective spaces beyond corank $p$7

Prior work (HMY) computed $p$8-homology of $p$9 completely only for corank below roughly $\pi_{2n}BU(r)$0, where the Hurewicz map is surjective. Here the authors push past this bound by ad hoc construction. A second splitting lemma shows that if $\pi_{2n}BU(r)$1 is $\pi_{2n}BU(r)$2-orientable and $\pi_{2n}BU(r)$3 has nonzero $\pi_{2n}BU(r)$4-Hurewicz image, then both top and bottom cells of $\pi_{2n}BU(r)$5 split (the top-cell splitting follows from Spanier–Whitehead duality and the relation $\pi_{2n}BU(r)$6). This yields $\pi_{2n}BU(r)$7-detected 2-torsion bundles on $\pi_{2n}BU(r)$8 of rank $\pi_{2n}BU(r)$9, and $\pi_*^s CP_r^n$0-detected bundles of ranks $\pi_*^s CP_r^n$1 and $\pi_*^s CP_r^n$2 on projective spaces of dimensions one less than the corresponding spheres.

For odd primes the rigid top-and-bottom splitting no longer aligns with the available class degrees, so the argument uses partial splittings into Adams summands. For $\pi_*^s CP_r^n$3, the composite factors through the cofiber $\pi_*^s CP_r^n$4 appearing as the top summand of $\pi_*^s CP_r^n$5; nonvanishing then hinges on the fact that the $\pi_*^s CP_r^n$6-Hurewicz image of $\pi_*^s CP_r^n$7 is not divisible by $\pi_*^s CP_r^n$8—established via Nave's homotopy fixed point spectral sequence differentials ($\pi_*^s CP_r^n$9 supports a nonzero $KO$00-differential) together with Belmont–Shimomura's charts. The analogous $KO$01 construction works cleanly only for $KO$02: the congruence forcing the top summand to be a suspension of $KO$03 fails for general $KO$04. Notably, the authors do not prove that pullbacks built from $KO$05, $KO$06, are trivial—they simply cannot detect them by this method.

Finally, since each sphere bundle pulls back under $KO$07 to a stably trivial bundle detected in the same theory, all sphere results propagate to projective spaces, producing nontrivial stably trivial bundles there as well.

Limitations and open questions

Several limitations are explicit. First, the method is purely a detection tool: it establishes existence of torsion but not divisibility or group structure. Second, for $KO$08 the finite known Hurewicz image bounds the produced families; whether $KO$09 has infinite Hurewicz image is open and affects how far these constructions extend. Third, the $KO$10 case for $KO$11 on projective spaces is undetected—the numerology obstructing the argument may reflect genuine triviality or merely a defect of the technique, and the paper leaves this question open. Fourth, the orientability conjecture $KO$12 of Bhattacharya–Chatham remains unproven in general, so improvements in orientability would directly enlarge the range of applicability.

Conclusion

The paper demonstrates that chromatic and equivariant cohomology theories with controlled orientation orders can systematically detect high-corank torsion in $KO$13 and produce nontrivial stably trivial bundles on $KO$14 and $KO$15 well beyond the corank ranges accessible to uniform computation. Its contributions are a pair of splitting lemmas converting orientability plus Hurewicz-image data into bundle detection, and explicit applications at primes 2 and 3 yielding infinite families, with finite but uniform families for all larger primes.