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For space pairs $(X_1,A_1)$ and $(X_2,A_2)$ ,is there a groups M,such that $\pi_{k}(X_1\vee X_2,A_1\vee A_2)\cong \pi_k(X_1,A_1)\oplus M$?

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$\pi_3(S_2 \vee S_2) = \mathbb Z^3\neq \mathbb Z^2 = \pi_3(S_2) \oplus \pi_2(S_2)$. I can't imagine that the relative homotopy groups would make it more well-behaved. –  Will Sawin Oct 6 '12 at 3:55
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up vote 6 down vote accepted

The pair $(X_1,A_1)$ is a retract of the pair $(X_1\vee X_2,A_1\vee A_2)$, so the map $\pi_k(X_1,A_1)\to \pi_k(X_1\vee X_2,A_1\vee A_2)$ is a split injection for all $k$. If $k\ge 3$ then the groups are abelian, and your result follows with $M$ the cokernel.

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There are some cases when one can give complete information. We know that the relative homotopy group $\pi_k(X_i,A_i)$ is a $\pi_1(A_i)$-module for $k >2$, crossed $\pi_1(A_i)$-module if $k=2$.

Suppose that the spaces are connected, well pointed, and the pairs $(X_1,A_1), (X_2,A_2)$ are $(k-1)$-connected. This result is a special case of the Higher Homotopy Seifert-van Kampen Theorem (HHSvKT) stated and proved in the book ``Nonabelian Algebraic Topology", that we have a pushout diagram

$$\matrix{(1,1)&\to&(\pi_1(A_1),\pi_k(X_1,A_1)) \cr \downarrow & & \downarrow \cr (\pi_1(A_2),\pi_k(X_2,A_2)) &\to & (\pi_1(A_1 \vee A_2),\pi_k(X_1 \vee X_2,A_1 \vee A_2))}$$ which is of modules if $k >2$, crossed modules if $k=2$. Here we write a $G$-module $M$ as a pair $(G,M)$, and we also use that notation in the crossed case though that involves also a morphism $\mu: M \to G$. Thus the result of the pushout is a kind of free product. This pushout diagram, or ``free product'', yields an explicit description of the resulting relative group. For convenience let us write the pushout as

$$\matrix{(1,1)& \to & (G_1,M_1) \cr \downarrow & & \downarrow \cr (G_2,M_2) & \to &(G_1*G_2, M_1 \vee M_2)} $$

Let $i; G_1 \to G_1 * G_2, i_2; G_2 \to G_1 * G_2$ be the canonical morphisms. We can form the induced $G_1 * G_2$-modules $i_*(M_1), j_*(M_2)$. Then for $k>2$, $M_1 \vee M_2 \cong i_*(M_1) \oplus j_*(M_2)$. For $k=2$ we have to use induced crossed modules and the coproduct of crossed $G_1 * G_2$-modules, but the relevant constructions are of course more complicated.

For the case $k>2$ there is presumably a proof using covering spaces and homology, but the case $k=2$, being nonabelian, looks inaccessible to those methods, and being a part of low dimensional topology, might also be relevant to geometric group theory. (The main theorem referred to dates from 1981, in a paper entitled ``Colimit theorems for relative homotopy groups'', JPAA. )

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