Let $\mathbb{D}_n^d$ be the $d$-dimensional unit ball with $n$ punctures. I am interested in the groups of orientation-preserving diffeomorphisms $Diff(\mathbb{D}_n^d)$ that fix the punctures. In particular I want to know what we know about all fundamental groups of this space $$\pi_k(Diff(\mathbb{D}_n^d))$$
E.g. I would assume it is zero for $k\geq d$? Also for $k<d-1$? Or: Is there a good presentation via action on the cohomology $H^k(\mathbb{D}_n^d)$ or even the homotopy groups (which I know are not known).

The example $d=2$ is the braid group in $n$ strands, which acts on $\mathbb{Z}^n$ (cohomology) and moreover on the free group in $n$ generators. This is what I have in mind and want to understand for higher $d$.

Thanks for your help! And apologies, that my knowledge in algebraic topology is limited. Even more I appreachiate any hint you may have! Greetings, Simon

  • $\begingroup$ You mean homotopy groups, not fundamental groups. $\endgroup$ – Ben McKay Feb 23 '19 at 8:45

Let me assume that you are interested in diffeomorphisms which also fix the boundary of the ball. If $F_n(D^d)$ denotes the space of $n$ distinct ordered points in $D^d$, then there is a fibration sequence $$Diff(D^d_n) \to Diff(D^d) \to F_n(D^d),$$ where the rightmost map is the orbit map given by acting on a fixed $\xi \in F_n(D^d)$. On the other hand, the leftmost map is a split surjection up to homotopy, by choosing a small $d$-ball which misses the points $\xi$. Thus there are split short exact sequences $$0 \to \pi_{k+1}(F_n(D^d)) \to \pi_k(Diff(D^d_n)) \to \pi_k(Diff(D^k)) \to 0.$$ One therefore has to analyse the two outer groups.

1) The space $F_n(D^d)$ fits into fibration sequences $$\vee^{n-1} S^{d-1} \simeq D^d \setminus \{n-1 \text{ points}\} \to F_n(D^d) \to F_{n-1}(D^d),$$ where the right-hand map forgets the last point. This fibration has a section by adding an $n$th point near the boundary of the disc $D^d$. Thus by induction $$\pi_k(F_n(D^d)) = \pi_k((\vee^{n-1} S^{d-1}) \times (\vee^{n-2} S^{d-1}) \times \cdots \times (S^{d-1})).$$

2) Apart from $d \leq 3$, where $Diff(D^d) \simeq *$ by theorems of Smale and of Hatcher, the groups $\pi_k(Diff(D^d))$ are far more mysterious, and are closely connected with subtle questions in differential topology. In high dimensions mostly only rational information is available, in the form of the calculation of Farrell and Hsiang that $$\pi_k(Diff(D^{d}))\otimes \mathbb{Q}=\begin{cases} \mathbb{Q} & \text{ if $d$ is odd and } k = 3 \mod 4\\ 0 & \text{ else} \end{cases}$$ as long as $k$ is in the pseudoisotopy stable range for $d$, which is approximately $k \leq d/3$.


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