Failure of excision for $\pi_2$

Solution 1:

Consider the excisive triad $(\Sigma X; CX,CX)$ induced by the decomposition of the reduced suspension of space $X$ as two reduced cones of $X$ attached along $X$. Notice that $CX\cap CX\cong X$. Now the long exact sequences of the pairs $(\Sigma X, CX)$ and $(CX, X)$ give the isomorphisms $\pi_2(\Sigma X, CX)\cong \pi_2(\Sigma X)$ and $\pi_2(CX,X)\cong \pi_1(X)$, since the reduced cone $CX$ is always contractible.

Therefore, we only need to provide an example of failure of the isomorphism : $\pi_2(\Sigma X)\cong\pi_1(X)$. Various examples can be given only by noticing that in general $\pi_1(X)$ is not abelian whereas $\pi_2(\Sigma X)$ is always abelian.

Thus, any based space with a non-abelian fundamental group gives you an example. The first one which comes to my mind is : $S^1\vee S^1$.

Solution 2:

Fortunately, and covering the points already made, there is a fairly complete answer to the determination of the excision map

$$\varepsilon_2 : \pi_2(B,C) \to \pi_2(X,A)$$ when $X=A \cup B, C = A \cap B$ where the base point lies in $C$, and some other conditions, such as $A,B$ are open; we also need connectivity conditions. We say $(B,C)$ is connected if $B,C$ are path connected and the morphism $\pi_1(C) \to \pi_1(B)$ is surjective. Under all these conditions, the excision result is that $(X,A)$ is connected, and that the morphism $\varepsilon_2$ is entirely determined by the morphism $\lambda: \pi_1(C) \to \pi_1(A)$ induced by the inclusion $C \to A$.

To give more detail, we need to recognise, with Henry Whitehead, 1946, that the boundary map $\delta: \pi_2(X,A) \to \pi_1(A)$ has the structure of crossed module.

A morphism $\mu: M \to P $ of groups is called a crossed module if there is given an action of the group $P$ on the group $M$, written $(m,p) \mapsto m^p$, such that the following two rules hold for all $p \in P, m,n \in M$:

CM$1$) $\delta(m^p)= p^{-1}mp$;

CM$2$) $n^{-1}mn= m^{\mu n}$.

The second rule is crucial, for example to the homotopical applications.

Now suppose $\mu : M \to P$ is a crossed module and $\lambda: P \to Q$ is a morphism of groups. Then we can construct a new crossed module $\delta: \lambda_* M \to Q$ called the crossed module induced from $\mu: M \to P$ by $\lambda $, which comes with a crossed module morphism $\lambda': M \to \lambda_*M \;$ and which with $\lambda $ satisfies a nice universal property for morphisms of crossed modules.

Then we have:

Homotopical Excision in Dimension $2$: Under the above conditions at the start of this account, $$\pi_2(X,A) \cong \lambda_* \pi_2(B,C).$$

An important point is that this result is about nonabelian structures and so not apparently deducible from standard homological tools.

An expository account of the background to this was published as the first article, available here, in the journal HHA, but the first statement and proof were in

R. Brown and P.J. Higgins, "On the connection between the second relative homotopy groups of some related spaces", Proc. London Math. Soc. (3) 36 (1978) 193-212.

which deduces it from a much more general $2$-dimensional van Kampen type Theorem stated and proved there. A full account is also in the new book on Nonabelian Algebraic Topology (EMS Tract Vol 15, 2011) advertised here. This Excision result is Theorem 5.4.1.

This leads to ideas for calculating induced crossed modules in homotopically relevant situations. See for example

R. Brown and C.D. Wensley, "Computation and homotopical applications of induced crossed modules", J. Symbolic Computation 35 (2003) 59-72.

which gives some computer based calculations.

The basic philosophy is that a $2$-d van Kampen Theorem allows the computation of some homotopy $2$-types, and from this, the calculation of some second homotopy groups, which are, after all, but a pale shadow of the $2$-type.

Addition 30 Dec: The universal property of the induced crossed module can be stated as that the following diagram of morphisms of crossed modules

$$ \begin{matrix} 1\to P & \to & 1 \to Q \\ \downarrow & & \downarrow \\ M \to P & \to & \lambda_* M \to Q \end{matrix} $$

is a pushout of crossed modules. This should make clear the connection with a van Kampen type theorem, dealing more generally with pushouts of crossed modules involving second relative homotopy groups. This Excision result also implies Whitehead's subtle theorem (in Combinatorial Homotopy II, 1949) that $\pi_2(A \cup_{f_i} \{e^2_i\}, A,a) \to \pi_1(A,a)$ is for connected $A$ the free crossed module on the characteristic maps $f_i$ of the $2$-cells, as well as the result given by mph on $\pi_2(\Sigma X)$ but without using homological information.

Edit 05/01/14: In particular, if $Q=1$ then $\lambda_* M$ is $M$ factored by the action of $P$. This implies the Relative Hurewicz Theorem in dimension $2$.

Edit Dec 31: Another interesting aspect of this result is that the failure of excision is usually related to the work of Blakers and Massey on triad homotopy groups, which gives an exact sequence

$$\to \pi_3(X,A) \to \pi_3(X;A,B) \to \pi_2(B,C) \to \pi_2(X,A) \to \pi_2(X;A,B) \to . $$

But it is not clear to me how this sequence can yield the above Excision Theorem, and Blakers and Massey's results on determining the first non vanishing triad group do not deal with the non simply connected case.

Edit: Jan 7 2014: I should add that in the paper R. Brown and J.-L.Loday, "Excision homotopique en basse dimension'', C.R. Acad. Sci. Paris S\'er. I 298 (1984) 353-356, and available here, we announce under the above conditions an exact sequence

$$\pi_3(B,C)\to \pi_3(X,A) \to \pi_2(A,C) \otimes \pi_2(B,C) \to \pi_2(B,C) \to \pi_2(X,A) $$ where $\otimes$ denotes a "nonabelian tensor product" of two groups which act on each other, basically via a determination of $\pi_3(X;A,B)$ by a van Kampen type theorem.