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Functorial seminorms on singular homology and (in)flexible manifolds. (English) Zbl 1391.57008

Summary: A functorial seminorm on singular homology is a collection of seminorms on the singular homology groups of spaces such that continuous maps between spaces induce norm-decreasing maps in homology. Functorial seminorms 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 seminorms with interesting properties. In particular, we answer a question of M. Gromov from [Metric structures for Riemannian and non-Riemannian spaces. Transl. from the French by Sean Michael Bates. With appendices by M. Katz, P. Pansu, and S. Semmes. Edited by J. LaFontaine and P. Pansu. Boston, MA: Birkhäuser (1999; Zbl 0953.53002)] by providing a functorial seminorm 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 M. Arkowitz and G. Lupton [Math. Z. 235, No. 3, 525–539 (2000; Zbl 0968.55005)]; we extend their methods to produce a wide variety of such manifolds.

MSC:

57N65 Algebraic topology of manifolds
55N10 Singular homology and cohomology theory
55N35 Other homology theories in algebraic topology
55P62 Rational homotopy theory
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