Scalar Curvature Question [?87]: but there are no apparent examples (if any) where these inequalities are strict

Definitions and mathematical statement

The source asks for examples, if any, in which certain K-area inequalities are strict. The phrase "these inequalities" precedes their missing antecedent, so the exact strictness question is not completely specified. The following supplied passage discusses Fredholm K-area and covariant push-forwards, but does not define the invariant itself.

For a covering $f:X\to Y$ of oriented Riemannian manifolds and a Hilbert bundle $L\to X$, it describes the direct-image fiber as \[ (f_*L)_y=\widehat{\bigoplus}_{x\in f^{-1}(y)}L_x. \] The Hilbert direct sum is the reading supported by its example: for the trivial covering $Y\times\Sigma\to Y$, with $\Sigma$ countable and $L$ the trivial line bundle, the image has fiber $\ell^2(\Sigma)$. It asserts that a Fredholm homomorphism $\Phi:L_1\to L_2$ invertible at infinity yields a corresponding Fredholm homomorphism of direct images, also invertible at infinity, with preserved curvature norms. The analytic uniformity and metric-compatibility hypotheses needed for an infinite covering are not stated.

It then displays an inequality for fundamental classes of manifolds or pseudomanifolds, \[ \operatorname{Fredholm\ K\!\text{-}area}[X_1] \geq\operatorname{Fredholm\ K\!\text{-}area}[X_2], \] but does not identify $X_1,X_2$ with the covering source and target $X,Y$. Thus no direction in terms of $f:X\to Y$ is inferred from the inconsistent labels.

For a fibration $f:X\to Y$ with positive-dimensional fibers $Z_y$, the passage gives continuous fiberwise Riemannian structures and a tangent-associated bundle $\Theta(Z_y)$, using \[ M_y=L^2(Z_y;L|_{Z_y}\otimes\Theta(Z_y)) \] as the push-forward fiber. With $Z_y$ closed, even-dimensional, and spin, and with fiberwise unitary connections continuous in $y$, twisted Dirac operators supply Fredholm data when $L$ has finite rank. The source permits noncompact bases and unbounded operators, unlike a usual compact-base bounded-operator setup; domains, continuity, and infinity conditions remain to be specified. The text stops in the spinor discussion. These constructions and qualifications are retained as source background rather than a fully proved functoriality theorem.

Short English statement

Find strictness examples for the source's K-area inequalities and clarify the Fredholm push-forward framework for coverings and spin fibrations. The relevant antecedent, invariant definition, analytic hypotheses, and source/target labels are incomplete.

Sources

• Misha Gromov, 101 Questions, Problems and Conjectures around Scalar Curvature (2017), printed tag [?87], occurrence 88, PDF page 85, including operator-convention footnotes in the supplied extract, https://www.ihes.fr/~gromov/wp-content/uploads/2018/08/101-problemsOct1-2017.pdf; accessed 2026-07-29, PDF-text-normalized extraction. The provenance describes the public author PDF as incomplete.
• Local registry data/unsolvedmath/problems.json, permanent UnsolvedMath ID $6700085$, source code catalogue entry, Scalar Curvature Question [?87]. Dataset attribution: Ulam AI and the UnsolvedMath Contributors.