Self-maps of simply connected manifolds
Contents |
1 Question
Let us call an oriented closed connected manifold flexible if it admits a self-map that has non-trivial degree (i.e., degree not equal to 1, 0, or -1).
Question 1.1. Do there exist closed simply connected manifolds (of non-zero dimension) that are not flexible?
Remark 1.2. In the following, for simplicity, we implicitly assume that all manifolds are of non-zero dimension.
2 Examples and partial answers
2.1 Examples of flexible manifolds
- Of course, all spheres are flexible.
- All odd-dimensional real projective spaces are flexible; all complex projective spaces are flexible.
- Products of flexible manifolds with oriented closed connected manifolds are flexible; in particular, tori are flexible.
- All closed simply connected 3-manifolds are flexible.
- All closed simply connected 4-manifolds are flexible [Duan&Wang2004, Corollary 2].
- ...
2.2 Examples of manifolds that are not flexible
Notice that there are many oriented closed connected manifolds that are not flexible.
Example 2.1.
- All oriented closed connected manifolds with non-zero simplicial volume are not flexible (because the simplicial volume is functorial.
- This includes, for instance, oriented closed connected manifolds of non-positive sectional curvature. (More examples and explanations of these facts can be found on the page on simplicial volume.)
However, by a theorem of Gromov, the simplicial volume of closed simply connected manifolds is always zero. So the simplicial volume cannot be used to discover closed simply connected manifolds that are not flexible.
3 Solution
This problem is was in fact Solved in [Arkowitz&Lupton2000, Examples 5.1 & 5.2]. There they give examples of 1-connected rational Poincare differential graded algebras, and of dimensions 208 and 228. Both of these algebras have a finite set of homotopy classes of self maps and so are inflexible They also state that the alebras and can be realised by simply connected manifolds and . It follows that the manifolds and are inflexible.
The realisation of and $\mathcal{M}_2 relies upon a theorem proven independently by Barge and Sullivan. A special case of this theorem is as follows
Theorem 3.1 c.f. [Barge1976, Theorem 1] and [Sullivan1977, Theorem 13.2]. Let be a simply connected differential rational Poincare algebra of dimension . If either
- or
- or and the intersection form of represents the trivial element of , the Witt group of ,
then there is a close simply connected PL manifold with rational homotopy type given by . If then may be given a smooth structure and if there can be smoothed except possible at one point.
Remark 3.2. Note that [Arkowitz&Lupton2000] simply state that the theorem of Barge-Sullivan applies and do not give arguments that or satisfy the hypotheses of Barge. See the discussion page for more information.
4 Further discussion
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5 References
- [Arkowitz&Lupton2000] M. Arkowitz and G. Lupton, Rational obstruction theory and rational homotopy sets, Math. Z. 235 (2000), no.3, 525–539. MR1800210 (2001h:55012) Zbl 0968.55005
- [Barge1976] J. Barge, Structures différentiables sur les types d'homotopie rationnelle simplement connexes, Ann. Sci. École Norm. Sup. (4) 9 (1976), no.4, 469–501. MR0440574 (55 #13448) Zbl 0348.57016
- [Duan&Wang2004] H. B. Duan and S. C. Wang, Non-zero degree maps between -manifolds, Acta Math. Sin. (Engl. Ser.) 20 (2004), no.1, 1–14. MR2056551 (2005b:57051) Zbl 1060.57018
- [Sullivan1977] D. Sullivan, Infinitesimal computations in topology, Inst. Hautes Études Sci. Publ. Math. (1977), no.47, 269–331 (1978). MR0646078 (58 #31119) Zbl 0374.57002