# 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 was in fact solved by Arkowitz and Lupton [Arkowitz&Lupton2000, Examples 5.1 & 5.2]: There do exist inflexible closed simply connected manifolds.

Arkowitz and Lupton give examples of simply connected rational Poincaré differential graded algebras, $\mathcal{M}_1$$\newcommand{\Z}{\mathbb{Z}} \newcommand{\R}{\mathbb{R}} \newcommand{\C}{\mathbb{C}} \newcommand{\N}{\mathbb{N}} \newcommand{\Q}{\mathbb{Q}} \newcommand{\F}{\mathbb{F}} \newcommand{\bZ}{\mathbb{Z}} \newcommand{\bR}{\mathbb{R}} \newcommand{\bC}{\mathbb{C}} \newcommand{\bH}{\mathbb{H}} \newcommand{\bQ}{\mathbb{Q}} \newcommand{\bF}{\mathbb{F}} \newcommand{\bN}{\mathbb{N}} \DeclareMathOperator\id{id} % identity map \DeclareMathOperator\Sq{Sq} % Steenrod squares \DeclareMathOperator\Homeo{Homeo} % group of homeomorphisms of a topoloical space \DeclareMathOperator\Diff{Diff} % group of diffeomorphisms of a smooth manifold \DeclareMathOperator\SDiff{SDiff} % diffeomorphism under some constraint \DeclareMathOperator\Hom{Hom} % homomrphism group \DeclareMathOperator\End{End} % endomorphism group \DeclareMathOperator\Aut{Aut} % automorphism group \DeclareMathOperator\Inn{Inn} % inner automorphisms \DeclareMathOperator\Out{Out} % outer automorphism group \DeclareMathOperator\vol{vol} % volume \newcommand{\GL}{\text{GL}} % general linear group \newcommand{\SL}{\text{SL}} % special linear group \newcommand{\SO}{\text{SO}} % special orthogonal group \newcommand{\O}{\text{O}} % orthogonal group \newcommand{\SU}{\text{SU}} % special unitary group \newcommand{\Spin}{\text{Spin}} % Spin group \newcommand{\RP}{\Rr\mathrm P} % real projective space \newcommand{\CP}{\Cc\mathrm P} % complex projective space \newcommand{\HP}{\Hh\mathrm P} % quaternionic projective space \newcommand{\Top}{\mathrm{Top}} % topological category \newcommand{\PL}{\mathrm{PL}} % piecewise linear category \newcommand{\Cat}{\mathrm{Cat}} % any category \newcommand{\KS}{\text{KS}} % Kirby-Siebenmann class \newcommand{\Hud}{\text{Hud}} % Hudson torus \newcommand{\Ker}{\text{Ker}} % Kernel \newcommand{\underbar}{\underline} %Classifying Spaces for Families of Subgroups \newcommand{\textup}{\text} \newcommand{\sp}{^}\mathcal{M}_1$ and $\mathcal{M}_2$$\mathcal{M}_2$ of dimensions 208 and 228. Both of these algebras have a finite set of homotopy classes of self-maps and so the corresponding rational Poincaré complexes are inflexible. They also indicate why the algebras $\mathcal{M}_1$$\mathcal{M}_1$ and $\mathcal{M}_2$$\mathcal{M}_2$ can be realised by closed simply connected manifolds $M_1$$M_1$ and $M_2$$M_2$. It follows that the manifolds $M_1$$M_1$ and $M_2$$M_2$ are inflexible.

The realisation of $\mathcal{M}_1$$\mathcal{M}_1$ and $\mathcal{M}_2$$\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 [Barge1976, Theorem 1][Sullivan1977, Theorem 13.2]. Let $\mathcal{M}$$\mathcal{M}$ be a simply connected rational Poincaré differential graded algebra of dimension n. If either

• $n \neq 4k$$n \neq 4k$ or
• $n = 4k$$n = 4k$ and the intersection form of $\mathcal{M}$$\mathcal{M}$ represents the trivial element of $W_0(\Qq)$$W_0(\Qq)$, the Witt group of $\Qq$$\Qq$,

then there is a closed simply connected smooth manifold $M$$M$ with rational homotopy type given by $\mathcal{M}$$\mathcal{M}$.

Remark 3.2. Note that Arkowitz and Lupton [Arkowitz&Lupton2000] state that the theorem of Barge-Sullivan applies and do not give detailed arguments why $\mathcal{M}_1$$\mathcal{M}_1$ or $\mathcal{M}_2$$\mathcal{M}_2$ satisfy the hypotheses of Barge-Sullivan. See the discussion page for more information.