Functorial seminorms on singular homology and (in)flexible manifolds

Abstract

ABSTRACT. A functorial semi-norm on singular homology is a collection of semi-norms on the singular homology groups of spaces such that continuous maps between spaces induce norm-decreasing maps in homology. Functorial semi-norms can be used to give constraints on the possible mapping degrees of maps between oriented manifolds.In this paper, we use information about the degrees of maps between manifolds to construct new functorial semi-norms with interesting properties. In particular, we answer a question of Gromov by providing a functorial semi-norm that takes finite positive values on homology classes of certain simply connected spaces. Our construction relies on the existence of simply connected manifolds that are inflexible in the sense that all their self-maps have degree −1, 0, or 1. The existence of such manifolds was first established by Arkowitz and Lupton; we extend their methods to produce a wide variety of such manifolds.

Exotic Finite Functorial Semi-Norms on Singular Homology

“…While the general classification of functorial semi-norms on singular homology is out of reach, one can ask for the role of the ℓ 1 -semi-norm among all finite functorial semi-norms [1,Question 5.8]. A simple rescaling manipulation shows that not all finite functorial semi-norms on singular homology are dominated by a multiple of the ℓ 1 -semi-norm [1,Section 5]. Relaxing the domination condition, we introduce the following relation between functorial semi-norms: Definition 1.1 (carriers of functorial semi-norms).…”

Branched coverings of simply connected manifolds

“…X m −→ M k ); recall that S 5 admits self-maps of any degree and it is minimal for the domination relation. In particular, we deduce, without using formality (see [4]), that the set of self-mapping degrees for every simply connected five-manifold is infinite.…”

“…N = # k T n−1 for some k ≥ 0), or more generally, by a product of type S m × N. This is a considerably stronger question than that of [15], however, it seems less likely to be true. For instance, simply connected manifolds which admit self-maps of absolute degree at most one might not be dominated by products (at least) of type S m × N. Such manifolds exist and are called inflexible; see for example [4]. In this direction, and since the set of self-mapping degrees of S m × N is infinite, we ask the following (at least in the simply connected case): Are there examples of inflexible manifolds that are dominated by flexible ones (i.e.…”

Homotopically rigid Sullivan algebras and their applications

“…Inflexible manifolds naturally appear within the framework of functorial seminorms on singular homology developed by Gromov [13,12] and derived degree theorems (e.g. [6,Remark 2.6]): let M be a closed, oriented and connected manifold with fundamental class c M . If there exists a functorial seminorm on singular homology | · | such that |c M | = 0, then M is inflexible.…”

“…But ℓ 1 -seminorm is trivial on simply connected manifolds [12, Section 3.1], which led Gromov to raise the question of whether every functorial seminorm on singular homology is trivial on all simply connected spaces [13,Remark (b) in 5.35]. This question is solved in the negative in [6] by constructing functorial seminorms associated to simply connected inflexible manifolds. Therefore, inflexible manifolds are extraordinary objects and still not many examples are known.…”