Ordinary Versus Immersive Simplicial Volume

Definitions and mathematical statement

A connected manifold $M$ is aspherical if its universal cover is contractible, equivalently if $\pi_k(M)=0$ for every integer $k\geq 2$.

The supplied question asks whether \[ \lVert M\rVert=\lVert M\rVert_{\mathrm{imm}} \] for every aspherical manifold $M$. The record does not specify compactness, boundary, orientability, smoothness, coefficients, or the precise meaning of an immersive simplex. These omissions prevent a unique fully specified formulation. The following gives a conditional formulation under explicit conventional hypotheses; these hypotheses and the immersion convention require confirmation against the source.

Let $n\geq 0$, and suppose that $M$ is a closed, connected, oriented, smooth $n$-manifold. Here closed means compact without boundary. Write \[ \Delta^n=\left\{(t_0,\ldots,t_n)\in{\mathbb{R}}^{n+1}:t_i\geq 0,\ \textstyle\sum_{i=0}^n t_i=1\right\}. \] A real singular fundamental cycle is a finite chain \[ z=\sum_{j=1}^r a_j\sigma_j\in C_n(M;{\mathbb{R}}),\qquad \partial z=0,\qquad [z]=[M]_{{\mathbb{R}}}\in H_n(M;{\mathbb{R}}), \] where the $\sigma_j:\Delta^n\to M$ are distinct continuous singular simplices and the $a_j$ are nonzero real coefficients. Set $\lVert z\rVert_1=\sum_{j=1}^r|a_j|$. The ordinary simplicial volume is \[ \lVert M\rVert =\inf\left\{\lVert z\rVert_1:z\text{ is a real singular fundamental cycle of }M\right\}. \] For a precise provisional immersion convention, call $\sigma:\Delta^n\to M$ immersive if it extends to a smooth map $\widetilde\sigma:U\to M$, where $U$ is an open neighborhood of $\Delta^n$ in its affine span, and \[ \operatorname{rank}(d\widetilde\sigma_x)=n \quad\text{for every }x\in\Delta^n. \] In particular, this convention requires full rank also at boundary points of the simplex. Define \[ \lVert M\rVert_{\mathrm{imm}} =\inf\left\{\lVert z\rVert_1:z\text{ is a real singular fundamental cycle and every simplex occurring in }z\text{ is immersive}\right\}, \] with $\inf\varnothing=+\infty$.

Under these explicit provisional conventions, the question is: for every closed, connected, oriented, smooth aspherical manifold $M$, does $\lVert M\rVert=\lVert M\rVert_{\mathrm{imm}}$ hold? The supplied wording refers to arbitrary aspherical manifolds; it does not establish that this closed smooth interpretation captures its full intended scope. Formulations for manifolds with boundary, noncompact manifolds, or nonorientable manifolds require additional choices that are not supplied.

Short English statement

The question asks whether restricting fundamental cycles to immersive singular simplices leaves the simplicial volume of every aspherical manifold unchanged. A precise conditional formulation is given for closed, connected, oriented smooth manifolds with real coefficients. The intended manifold class and the immersion convention remain unresolved in the supplied record.

Sources

• Local UnsolvedMath registry, supplied complete source record: permanent ID $30001810$, title Ordinary Versus Immersive Simplicial Volume, exact source problem code catalogue entry, in the Oberwolfach Reports Open Problems collection. The record gives the proposed year as 2011 and names no individual proposer.
• Arithmetic Groups vs. Mapping Class Groups: Similarities, Analogies and Differences (2011), Oberwolfach Reports, pp. 1637--1708, DOI 10.4171/owr/2011/30. Supplied source URL and literature-source label "original Oberwolfach report": https://doi.org/10.4171/owr/2011/30.
• Supplied literature-source label "Bucher--immersive simplicial volume": https://ems.press/content/serial-article-files/46346. The supplied record gives no fuller bibliographic identification for this link.
• Literature assessment dated exactly 2026-08-21: the supplied curation labels the problem open, reports equality for several locally symmetric spaces via immersive straightening, and states that the report poses the question for arbitrary aspherical manifolds without a general resolution. This is supplied dated curation, not an independently verified assessment; the record explicitly requests independent verification of time-sensitive status and cited claims.
• The supplied statement status is corrected_verified; its verification note says that mathematical content was checked against the official Oberwolfach report and that notation, context, and question form were normalized without changing the source-level claim. This verification is reported from the registry and has not been independently repeated here.