Bounded cohomology and simplicial volumeSemester
WiSe 2019 / 20
Type of course (Veranstaltungsart)
Bounded cohomology was first defined for groups in the seventies by Johnson and Trauber. Later, in his pioneering paper "Volume and bounded cohomology" (1982) Gromov extended the notion of bounded cohomology to topological spaces. Gromov's approach is intended to apply all this theory to both the construction and the study of some invariants of manifolds. Among the invariants discussed by Gromov, it is of our interest the simplicial volume of manifolds. Roughly speaking, simplicial volume measures how difficult it is to describe a manifold in terms of singular simplices (with coefficients). One may interpret it as a generalization of Euler characteristic (also if this correspondence is very loose) via the following still open question:
- Gromov's question (1993): Is it true that in the case of oriented closed connected aspherical manifolds, the vanishing of the simplicial volume implies the vanishing of the Euler characteristic?
The aim of this course is to describe bounded cohomology of spaces and study how it is related with both bounded cohomology of groups and simplicial volume of manifolds. To this end, we develop the theory of bounded cohomology of spaces and we prove Gromov's Mapping Theorem which (in a simplified version) states that the bounded cohomology of spaces only depends on their fundamental group. For instance, since bounded cohomology (in positive degree and with real coefficients) vanishes in presence of amenable groups, the previous result implies that bounded cohomology of spaces with amenable fundamental groups always vanishes (in positive degree and with real coefficients).
As anticipated above among the applications of the study of bounded cohomology, we focus our attention on the simplicial volume. Simplicial volume is a homotopy invariant of manifolds which was first introduced by Gromov in his proof of Mostow rigidity for hyperbolic manifolds. This proof is achieved via Gromov-Thruston's Proportionality Principle, which states that the Riemannian volume of hyperbolic manifolds is proportional to the simplicial volume. Hence, the Riemannian volume of hyperbolic manifolds is also a homotopy invariant. One of the main goals of the course is to prove this striking result, which can be seen as a generalization in higher dimension of Gauss-Bonnet Theorem. Simplicial volume also provides nice obstructions to the existence of maps between manifolds of certain degrees. For instance, we prove that hyperbolic manifolds cannot admit a self-map of degree (in modulus) greater than or equal to 2. The crucial relation between bounded cohomology and simplicial volume appears in Gromov's Duality Principle. We prove this result which allows us to compute the simplicial volume in terms of bounded cohomology. Since the bounded cohomology of the trivial group is zero, we prove that the simplicial volume of simply connected manifolds vanishes. Finally, we study Gromov's question and we introduce some strategies to obtain a positive answer in some cases.
Recommended previous knowledge
All participants should have a firm background in Analysis I/II (in particular, basic point set topology,
e.g., as in Analysis II in WS 2011/12), in Linear Algebra I/II, and basic knowledge in group theory (as
covered in the lectures on Algebra).
Knowledge about the following topics is _not_ necessary, but helpful: manifolds (as in Analysis IV), basic
homological algebra (as in the last two weeks of Kommutative Algebra in SS 2018), algebraic topology (as
in WS 2018/19), group cohomology (as in SS 2019).
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Course work (Studienleistungen)
- Registration for course work/examination/ECTS: FlexNow
- Successful participation in the exercise classes: 50% of points and presentation of at least
Regelungen bei Studienbeginn vor WS 2015 / 16
- Oral exam: Duration: 30 minutes, Date: individual, re-exam: Date: individual
- O. g. Studienleistung und o. g. Prüfungsleistung; die Note ergibt sich aus der Prüfungsleistung
BV, MV, MGAGeo