16
$\begingroup$

This was sparked because I wanted to compute $\pi_2(Sym^2(\Sigma_2))$ via $Sym^2(\Sigma_2)\approx \mathbb{T}^4$# $\bar{\mathbb{C}P}^2$.
We know how to compute $\pi_1$ of $M$ # $N$ via van-Kampen's theorem. But what about higher homotopy groups? I looked in the literature and google without luck, and so I am wondering if no such procedure exists. Are there any results for calculating $\pi_n$ of connected sums?

There was mention of "higher van Kampen theorem"... has this actually been used to do such computations? I'd be interested in references if not just examples.

$\endgroup$
6
  • 1
    $\begingroup$ Perhaps ncatlab.org/nlab/show/higher+homotopy+van+Kampen+theorem ? $\endgroup$ Apr 6, 2012 at 6:47
  • 1
    $\begingroup$ Higher homotopy groups, higher van Kampen theorems. $\endgroup$ Apr 6, 2012 at 13:39
  • $\begingroup$ Sorry can this even be used in a connected-sum example? It seems just like "abstract theory". $\endgroup$ Apr 6, 2012 at 16:26
  • 6
    $\begingroup$ I don't know of any examples of people using these higher homotopy van kampen theorems to compute anything new. But if people can point out examples that would be great. $\endgroup$ Apr 6, 2012 at 19:31
  • 1
    $\begingroup$ In answer to Ryan Budney, one example in the joint paper Brown and Loday, Topology, 26 (1987), 311-334, was the calculation of the 3-type of $X= S(K(G,1))$ in terms of the nonabelian tensor square $G \otimes G$, for any group $G$. My bibliography on the nonabelian tensor product has 127 items (my name is on 7 of them), and [115] is on new calculations using GAP. Group theorists are interested because the commutator map $G \times G \to G$ is a biderivation and so factors through a morphism $\kappa:G \otimes G \to G$ whose kernel is $\pi_3 S(K(G,1))$. Abstract??!! $\endgroup$ Aug 9, 2013 at 13:51

3 Answers 3

34
$\begingroup$

The 2nd homotopy group of a connect sum is fairly reasonable to compute. $\pi_i X$ is isomorphic to $\pi_i \tilde X$ provided $i \geq 2$ and $\tilde X$ indicates any covering space of $X$, so we might as well take the universal cover. By the Hurewicz theorem, $\pi_2 \tilde X$ is isomorphic to $H_2 \tilde X$. In the case of a connect sum, the universal cover has a very nice description (take disjoint unions of the universal covers of the punctured manifolds and glue them together appropriately).

Since $\mathbb CP^2$ is simply connected this is a fairly easy thing to compute. The universal cover looks like $\mathbb R^4$ with a $\mathbb CP^2$ summand at every integer lattice point. So,

$$\pi_2 ((S^1)^4 \# \mathbb CP^2) \simeq \bigoplus_{\pi_1 T^4} \pi_2 \mathbb CP^2$$

i.e. a direct sum over $\mathbb Z^4$ of copies of the integers, i.e. $\mathbb Z[t_1^\pm, t_2^\pm, t_3^\pm, t_4^\pm]$ a laurent polynomial ring in four variables. $\pi_1$ acts by multiplication by units in the Laurent polynomial ring.

Higher homotopy groups in general can be fairly painful to compute but $\pi_2$ is usually quite reasonable, like this case.

$\endgroup$
5
  • 1
    $\begingroup$ Ah perfect for $\pi_2$! Is there a deep reason why we can't do similar things for higher homotopy? $\endgroup$ Apr 6, 2012 at 16:41
  • 2
    $\begingroup$ The anthropic principal response would be "homotopy groups are really useful, so if you could compute them easily, your life would become too easy". :) I don't have a deep reason to offer you, but if you take as an axiom that the homotopy groups of spheres are very complicated and there's mysterious patterns relating them, then since $S^n=\Sigma S^{n−1}$, there can't be a simple way to compute the homotopy groups of a CW-complex since that would force too obvious a symmetry into the homotopy-groups of spheres. Rational homotopy groups are much more manageable, but torsion is tricky. $\endgroup$ Apr 6, 2012 at 19:29
  • 2
    $\begingroup$ The general framework I'm using starts running out of steam but it is usable. $\pi_2$ we compute by going to the universal cover and computing $H_2$. Okay, so what about $\pi_3$? You'd kill $\pi_2$ via a fibration and compute $H_3$ of the total space of that fibration. Basically we're using the Whitehead tower (as described in Hatcher's algtop book) where you kill the low-dimensional homotopy groups successsively. If $\pi_2$ is free abelian you can kill it using a fibre bundle with fibre a product of circles (a generalized Hopf fibration). $\endgroup$ Apr 6, 2012 at 20:03
  • $\begingroup$ This is, in principle, the old Serre method for computing homotopy groups, isn't it? $\endgroup$ Apr 9, 2012 at 10:09
  • 1
    $\begingroup$ I thought Serre did a similar thing but usually the other way around, using a Postnikov system rather than the Whitehead tower. They're very similar ideas, though. I would have known the answer when I was a grad student! $\endgroup$ Apr 9, 2012 at 19:15
10
$\begingroup$

Here is something that's valid in the stable range.

If $M$ and $N$ are closed $n$-manifolds, there is a cofibration sequence $$ S^{n-1} \to M_0 \vee N_0 \to M\sharp N $$ where $M_0$ denotes the effect of deleting a point from $M$.

If $M$ and $N$ are $r$-connected, then so is the connected sum. The Blakers-Massey excision theorem then implies an exact sequence $$ \pi_k(S^{n-1}) \to \pi_k(M_0 \vee N_0) \to \pi_k(M\sharp N) \to \pi_{k-1}(S^{n-1}) \to \cdots $$ as long as $k \le n-2+r$.

Furthermore the map $M_0 \vee N_0 \to M_0 \times N_0$ is $(2r+1)$-connected, so if $k \le 2r$ we get $\pi_k(M_0 \vee N_0) = \pi_k(M) \oplus \pi_k(N)$.

Assembling this, we have an exact sequence $$ \pi_k(S^{n-1}) \to \pi_k(M) \oplus \pi_k(N) \to \pi_k(M\sharp N) \to \pi_{k-1}(S^{n-1}) \to \cdots $$ which is valid for $k \le 2r$, $r \le n-2$.

Added Later

I just realized one could simply note that the cofiber sequence gives a long exact sequence on stable homotopy $$ \pi_k^{st}(S^{n-1}) \to \pi_k^{st}(M_0) \oplus \pi_k^{st}(N_0) \to \pi_k^{st}(M\sharp N) \to \pi_{k-1}^{st}(S^{n-1}) \to \cdots $$ and then if $M$ and $N$ are $r$-connected with $k \le 2r$ and $r\le n-2$ we can use the Freudenthal suspension theorem to identify the stable groups with the corresponding unstable ones. This gives a more elementary argument.

Here's a special case: when $M$ and $N$ are framed, so is $M\sharp N$ and the connecting map in the exact sequence splits to give a splitting $$ \pi_k(M\sharp N) = \pi_k(M) \oplus \pi_k(N) \oplus \pi_{k-1}(S^{n-1}) $$ (assuming the constraints on $k,r$ and $n$).

$\endgroup$
4
  • $\begingroup$ @ John Klein: The higher van Kampen theorems can sometimes give more information on $K \cup L$ when the intersection $K \cap L$ is not simply connected. For the connected sum we essentially have $P- K \cap L$ is a solid ball. What about other constructions, e.g. $P= S^1 \times D^2$ ? or more complicated? Are those of interest? $\endgroup$ Aug 9, 2013 at 17:01
  • $\begingroup$ That should have been $P=K \cap L$. One example which worked well was when $P= K(G,1)$ for any group $G$ and $K,L$ are cones on $P$, so their union is the suspension of $P$. $\endgroup$ Aug 9, 2013 at 20:20
  • $\begingroup$ What is the map $S^{n-1} \to M_0\vee N_0$? I'm having a hard time understanding this cofibration. I know of the cofibration $S^{n-1} \to M\#N \to M\vee N$, but not the one you're using here. $\endgroup$ Sep 21, 2017 at 20:43
  • $\begingroup$ @MichaelAlbanese The map in question is given as follows: let $S^{n-1} \to M_0$ and$S^{n-1} \to N_0$ be the attaching maps for the top cells of $M$ and $N$. Now form the composition $S^{n-1} \to S^{n-1} \vee S^{n-1} \to M_0 \vee N_0$, where the first map in the composite is the pinch map. $\endgroup$
    – John Klein
    Sep 23, 2017 at 17:20
5
$\begingroup$

Just a comment on the role of the Higher Homotopy Seifert-van Kampen Theorems:

they should be regarded as an extra tool in algebraic topology. There are quite severe conditions on their applicability but when they apply they compute quite a lot. Just as the 1-dimensional theorem, in its groupoid formulation, is about calculating 1-types, so the 2-dim theorem is about computing 2-types, in the form of crossed modules (over groupoids). However computing the second homotopy group from this 2-type may not be straightforward. But then the situation is the same for the 1-dim theorem, as is evidenced by the complications of theorems such as the Kurosh subgroup theorem, which can be seen to be about the fundamental group(s) of a cover of a wedge of $K(G_i,1)$'s.

As a taster, based on the 2-d theorem, work with Chris Wensley enabled the computation of the crossed module representing the 2-type of the mapping cone of a map $Bf: BG \to BH$ induced by a morphism $f: G \to H$. Of course. the second homotopy group, even as a module over the fundamental group, is but a pale shadow of the 2-type. You can see some of this in our book (pdf available from my web page on the book).

R. Brown, P.J. Higgins, R. Sivera, Nonabelian algebraic topology: filtered spaces, crossed complexes, cubical homotopy groupoids, EMS Tracts in Mathematics Vol. 15, 703 pages. (August 2011).

November 8, 2013: As a taster, let $X$ be the homotopy pushout of the classifying spaces of the two maps of groups $P \to P/M, P \to P/N$ where $M,N$ are normal subgroups of the group $P$. The the homotopy 2-type of $X$ is determined by the crossed module $M \circ N \to P$, the coproduct of the two crossed $P$-modules, which is given by the pushout of crossed modules

$$\begin{matrix} (1 \to P) & \to & (N \to P) \cr \downarrow && \downarrow \cr (M \to P)& \to & (M \circ N \to P) \end{matrix} $$. It follows that $$ \pi_2(X) \cong (M \cap N)/ [M,N]. $$ (Of course we know $\pi_1 X$ by the 1-dimensional van Kampen Theorem.) This result is applied in Bardakov, Valery G; Mikhailov, Roman; Vershinin, Vladimir V.; Wu, Jie, "Brunnian braids on surfaces". Algebr. Geom. Topol. 12 (2012), no. 3, 1607–1648.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.