Nonvanishing Euler Classes in Topological Bordism

Definitions and mathematical statement

Let $p$ be an odd prime, $G$ a finite abelian $p$-group, and $V$ a finite-dimensional complex $G$-representation with no trivial direct summand. Equivalently, $V^G=0$. Since $G$ is abelian, it decomposes as a sum of nontrivial one-dimensional characters; faithfulness of the representation is not assumed, and $V^G=0$ does not assert that every nonidentity element acts without fixed vectors.

Let $BG=EG/G$ and let $\xi_V=EG\times_G V\to BG$ be the associated bundle, of complex rank $r$. In an oriented bordism cohomology theory $E$ in which this bundle has its specified Thom class $u_E(\xi_V)$, its Euler class is \[ e_E(V)=s^*u_E(\xi_V)\in E^{2r}(BG), \] where $s$ is the zero section. The question is whether \[ e_E(V)\ne0 \] in the topological bordism theory intended by the report, for every such $G$ and $V$.

The extract does not name the representing bordism spectrum, orientation convention or localization/completion. Oriented topological bordism, complex cobordism and unoriented bordism are different theories; one cannot replace the intended theory by ordinary cohomology or by a chosen alternative simply to interpret the Euler class. The displayed $E$ notation exposes this missing specification. If an orientation supporting the Whitney-sum formula is fixed, the character decomposition expresses the Euler class as the product of the corresponding line-bundle Euler classes, but nonvanishing of the factors alone does not prove nonvanishing of their product.

The literature assessment checked on 21 August 2026 says the fixed-point transfer theorem in the source passage does not answer this Euler-class question and treats it as unresolved. No stronger consequence of that transfer theorem is supplied. This is dated triage requiring independent verification.

Short English statement

For an odd-primary finite abelian group, must a complex representation with no trivial summand have nonzero Euler class in the report's topological bordism theory? The exact bordism theory and orientation must be specified; fixed-point transfer alone is not the supplied answer.

Sources

• Cohomology of Finite Groups: Interactions and Applications (2006), Oberwolfach Reports, DOI 10.4171/owr/2005/42, pp. 2375--2434, https://doi.org/10.4171/owr/2005/42; proposed year supplied as 2005.
• The sole literature source in the assessment checked 21 August 2026 is the original report, https://doi.org/10.4171/owr/2005/42. It distinguishes the Euler-class question from the fixed-point transfer theorem in the passage.
• Local UnsolvedMath registry, permanent ID $30000323$, exact source code catalogue entry, created 25 August 2026 and updated 16 September 2026. The clean statement is labelled corrected and verified, restoring the finite abelian odd-$p$ group and representation hypotheses.