A crossed module is a pair of groups $C$ and $G$, an action of $G$ on $C$, and a homomorphism $\partial: C \to G$ that satisfy

• $\partial(g\cdot c)=g(\partial c)g^{-1}$, and
• $cc'c^{-1}=(\partial c)\cdot c'$

Let $(X,A)$ be a pointed pair of spaces. Whitehead proved that, in the homotopy long exact sequence of the pair, $$ \pi_2(X,A) \stackrel{\partial}{\rightarrow} \pi_1(A) $$ is a crossed module. Simply put, my question is what does this give us, other than an extra bit of structure? Does knowing that this is true aid on calculation? Does it aid in distinguishing spaces? Does it give us something really cool that I haven't thought of? (Probably the answer to this one is "yes".)

A crossed module is a pair of groups $C$ and $G$, an action of $G$ on $C$, and a homomorphism $\partial: C \to G$ that satisfy

• $\partial(g\cdot c)=g(\partial c)g^{-1}$, and
• $cc'c^{-1}=(\partial c)\cdot c'$

Let $(X,A)$ be a pointed pair of spaces. Whitehead proved that, in the homotopy long exact sequence of the pair, $$ \pi_2(X,A) \stackrel{\partial}{\rightarrow} \pi_1(A) $$ is a crossed module. Simply put, my question is: what does this give us, other than an extra bit of structure? Does knowing that this is true aid on calculation? Does it aid in distinguishing spaces? Does it give us something really cool that I haven't thought of? (Probably the answer to this one is "yes".)

A crossed module is a pair of groups $C$ and $G$, an action of $G$ on $C$, and a homomorphism $\partial: C \to G$ that satisfy

• $\partial(g\cdot c)=g(\partial c)g^{-1}$, and
• $cc'c=(\partial c)\cdot c'$

Let $(X,A)$ be a pointed pair of spaces. Whitehead proved that, in the homotopy long exact sequence of the pair, $$ \pi_2(X,A) \stackrel{\partial}{\rightarrow} \pi_1(A) $$ is a crossed module. Simply put, my question is what does this give us, other than an extra bit of structure? Does knowing that this is true aid on calculation? Does it aid in distinguishing spaces? Does it give us something really cool that I haven't thought of? (Probably the answer to this one is "yes".)

A crossed module is a pair of groups $C$ and $G$, an action of $G$ on $C$, and a homomorphism $\partial: C \to G$ that satisfy

• $\partial(g\cdot c)=g(\partial c)g^{-1}$, and
• $cc'c^{-1}=(\partial c)\cdot c'$

Let $(X,A)$ be a pointed pair of spaces. Whitehead proved that, in the homotopy long exact sequence of the pair, $$ \pi_2(X,A) \stackrel{\partial}{\rightarrow} \pi_1(A) $$ is a crossed module. Simply put, my question is what does this give us, other than an extra bit of structure? Does knowing that this is true aid on calculation? Does it aid in distinguishing spaces? Does it give us something really cool that I haven't thought of? (Probably the answer to this one is "yes".)

A crossed module is a pair to groups $C$ and $G$, an action of $G$ on $C$, and a homomorphism $\partial: C \to G$ that satisfy

• $\partial(g\cdot c)=g(\partial c)g^{-1}$, and
• $cc'c=(\partial c)\cdot c'$

Let $(X,A)$ be a pointed pair of spaces. Whitehead proved that, in the homotopy long exact sequence of the pair, that $$ \pi_2(X,A) \stackrel{\partial}{\rightarrow} \pi_1(A) $$ is a crossed module Simply put, my question is what does this give us, other than an extra bit of structure? Does knowing that this is aid on calculation? Does it aid in distinguishing spaces? Does it give us something really cool that I haven't thought of? (Probably the answer to this one is "yes".)

A crossed module is a pair of groups $C$ and $G$, an action of $G$ on $C$, and a homomorphism $\partial: C \to G$ that satisfy

• $\partial(g\cdot c)=g(\partial c)g^{-1}$, and
• $cc'c=(\partial c)\cdot c'$

Let $(X,A)$ be a pointed pair of spaces. Whitehead proved that, in the homotopy long exact sequence of the pair, $$ \pi_2(X,A) \stackrel{\partial}{\rightarrow} \pi_1(A) $$ is a crossed module. Simply put, my question is what does this give us, other than an extra bit of structure? Does knowing that this is true aid on calculation? Does it aid in distinguishing spaces? Does it give us something really cool that I haven't thought of? (Probably the answer to this one is "yes".)