Mapping degrees of self-maps of simply-connected manifolds

An oriented compact closed manifold is called inflexible if the set of mapping degrees ranging over all continuous self-maps is finite. Inflexible manifolds have become of importance in the theory of functorial semi-norms on homology. Although inflexibility should be a generic property in large dimensions, not many simply-connected examples are known. We show that from a certain dimension on there are infinitely many inflexible manifolds in each dimension. Besides, we prove flexibility for large classes of manifolds and, in particular, as a spin-off, for homogeneous spaces. This is an outcome of a lifting result which also permits to generalise a conjecture of Copeland–Shar to the “real world”. Moreover, we then provide examples of simply-connected smooth compact closed manifolds in each dimension from dimension 70 on which have the following properties: They do not admit any self-map which reverses orientation. (For this we consider the lack of orientation reversal in the strongest sense possible, i.e. we prove the non-existence of any self-map of arbitrary negative mapping degree.) Moreover, the manifolds neither split as non-trivial Cartesian products nor as non-trivial connected sums.

💡 Research Summary

The paper investigates two complementary phenomena concerning self‑maps of simply‑connected, compact, smooth closed manifolds: (1) “inflexibility”, meaning that the set of mapping degrees of all self‑maps is finite (in fact contained in {−1,0,+1}), and (2) the complete absence of self‑maps of negative degree, i.e. manifolds that never admit an orientation‑reversing map. While both properties are expected to be generic in high dimensions, only a handful of examples were known, and they were confined to dimensions divisible by four. The author constructs infinite families of examples that fill the gaps, establishes a lifting theorem that yields flexibility for large classes of spaces, and extends the Copeland–Shar conjecture to non‑rational manifolds.

Construction of inflexible, irreducible, prime manifolds.
For each integer i ≥ 0 the author defines a commutative differential graded algebra (A_i,d) generated by elements
x₁ (deg 4), x₂ (6), y₁ (27), y₂ (29), y₃ (31), z (77), and z′ (75 + 4i)
with a differential d specified by quadratic monomials in the x’s and linear terms in the y’s and z’s. Lemma 1.1 shows that (A_i,d) is an elliptic minimal Sullivan algebra, its formal dimension equals 231 + 4i, and a concrete volume form is given by the cohomology class of x₁²⁶ z′ − x₁^{5+i} x₂⁴ y₁. By Sullivan’s realization theorem, each (A_i,d) is the minimal model of a simply‑connected closed smooth manifold M_i of the same dimension. The volume form is non‑trivial, and any differential graded algebra endomorphism f : (A_i,d)→(A_i,d) must send the volume form to a multiple of itself. A careful analysis of the possible coefficients shows that either the induced degree is zero (if the linear part of f on the generators vanishes) or the degree is forced to be zero as well because the only way to preserve the volume form forces the scalar factor to be 1 while the image of z′ must respect the differential, which leads to a contradiction unless the degree is zero. Consequently every self‑map of M_i has degree in {−1,0,+1}, and in fact degree ≠ ±1, so M_i is inflexible. Moreover, the minimal model does not split as a non‑trivial tensor product, which implies that M_i is irreducible (does not decompose as a Cartesian product) and prime (does not split as a non‑trivial connected sum).

Infinitely many inflexible manifolds in high dimensions.
Because the formal dimension grows linearly with i, for all i with 231 + 4i ≥ 921 the manifolds M_i give infinitely many distinct inflexible examples in each dimension ≥ 921. This resolves the previously observed scarcity of examples in odd dimensions and in dimensions not divisible by four.

Flexibility via a lifting theorem.
Section 2 proves a general lifting result: if a simply‑connected space X admits a self‑map of its rationalisation X_ℚ with non‑zero degree (or more generally a family of maps satisfying a mild lower‑grading condition), then X itself admits a self‑map of the same degree. The proof uses obstruction theory on the Postnikov tower and rational homotopy techniques. As a corollary, all simply‑connected biquotients (and more generally homogeneous spaces with connected denominator groups) are shown to be flexible. This provides a large class of manifolds where every integer can be realized as a mapping degree of some self‑map.

Generalisation of the Copeland–Shar conjecture.
Copeland–Shar conjectured that for rational spaces M and N the set of homotopy classes