Self-maps of simply connected manifolds

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 (almost) solved; the solution will be presented soon on this page.

4 Further discussion

...

5 References

• [Duan&Wang2004] H. B. Duan and S. C. Wang, Non-zero degree maps between $2n$$<!-- COMMENT: To achieve a unified layout, along with using the template below, please OBSERVE the following: besides, $...$ and $$...$$, you should use two environments: - For statements like Theorem, Lemma, Definition etc., use e.g. {{beginthm|Theorem 1|(Milnor)}} ... ... ... {{endthm}}. - For references, use e.g. {{cite|Milnor1958b}}. END OF COMMENT --> == Question == <wikitex>; 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). {{beginthm|Question}} Do there exist closed simply connected manifolds (of non-zero dimension) that are not flexible? {{endthm}} {{beginthm|Remark}} In the following, for simplicity, we implicitly assume that all manifolds are of non-zero dimension. {{endthm}} </wikitex> == Examples and partial answers == === Examples of flexible manifolds === <wikitex>; * 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 {{cite|Duan&Wang2004|Corollary&nbsp;2}}. * ... </wikitex> === Examples of manifolds that are not flexible === <wikitex>; Notice that there are many oriented closed connected manifolds that are not flexible. {{beginthm|Example}} * All oriented closed connected manifolds with non-zero [[simplicial volume]] are not flexible (because the simplicial volume is [[Simplicial volume#Functoriality and elementary examples|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]].) {{endthm}} 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. </wikitex> == Solution == <wikitex>; This problem is (almost) solved; the solution will be presented soon on this page. </wikitex> == Further discussion == <wikitex>; ... </wikitex> == References == {{#RefList:}} <!-- Please add further headings according to your needs. --> [[Category:Questions]] [[Category:Research questions]]2n$-manifolds, Acta Math. Sin. (Engl. Ser.) 20 (2004), no.1, 1–14. MR2056551 (2005b:57051) Zbl 1060.57018