8. Group object and Loop space

Group object and Loop space

Definition 8.1. Let $X,Y\in \underline{{\mathcal{T}}_{\star }}$ be two pointed spaces. A based homotopy between two based maps $f_0,f_1:X\to Y$ is a homotopy between $f_0$, $f_1$ relative to the base points. We denote $[X,Y{]}_0$ to be the based homotopy classes of based maps. We define the category $\underline{h{\mathcal{T}}_{\star }}$ by the quotient of $\underline{{\mathcal{T}}_{\star }}$ where $${\mathrm{Hom}}_{\underline{h{\mathcal{T}}_{\star }}}(X,Y)=[X,Y{]}_0.$$

Definition 8.2. Given $(X,x_0)\in \underline{{\mathcal{T}}_{\star }}$, we define the based loop space ${\Omega }_{x_0}X$ or simply $\Omega X$ by $$\Omega X={\mathrm{Map}}_{\star }(S^1,X).$$In the unpointed case, we define the free loop space $$\mathcal{L}X=\mathrm{Map}(S^1,X).$$

Our goal in this section is to explore some basic algebraic structures of based loop spaces.

Theorem 8.3. The based loop space $\Omega$ defines functors $$\Omega :\underline{{\mathcal{T}}_{\star }}\mapsto \underline{{\mathcal{T}}_{\star }},\Omega :\underline{h{\mathcal{T}}_{\star }}\mapsto \underline{h{\mathcal{T}}_{\star }}.$$

Proof. Let us first consider $\Omega :\underline{{\mathcal{T}}_{\star }}\mapsto \underline{{\mathcal{T}}_{\star }}$. This amounts to showing that given $f:X\to Y$, the induced map $$f_{\ast }:{\mathrm{Map}}_{\star }(S^1,X)\to {\mathrm{Map}}_{\star }(S^1,Y),\gamma \mapsto f\circ \gamma$$is continuous. This follows from Proposition 5.14 since this map is the same as $${\mathrm{Map}}_{\star }(S^1,X)\times \{f\}\to {\mathrm{Map}}_{\star }(S^1,Y).$$

Now, we consider $\Omega :\underline{h{\mathcal{T}}_{\star }}\mapsto \underline{h{\mathcal{T}}_{\star }}$. We need to show that if we have a homotopy realized by $F:X\times I\to Y$, then the induced maps $f_{\ast }$, $g_{\ast }:{\mathrm{Map}}_{\star }(S^1,X)\to {\mathrm{Map}}_{\star }(S^1,Y)$ are also homotopic. The required homotopy is given by $$\Omega F:\Omega X\times I\to \Omega Y,(\gamma ,t)\mapsto F(-,t)\circ \gamma .$$To see the continuity of $\Omega F$, we first use the Exponential Law to express $F$ equivalently as a continuous map $\tilde{F}:I\to {\mathrm{Map}}_{\star }(X,Y)$. Then $\Omega F$ is given by the composition $${\mathrm{Map}}_{\star }(S^1,X)\times I\stackrel{1\times \tilde{F}}{\to }{\mathrm{Map}}_{\star }(S^1,X)\times {\mathrm{Map}}_{\star }(X,Y)\to {\mathrm{Map}}_{\star }(S^1,Y),$$which is continuous by Proposition 5.14.

$\square$

Definition 8.4. Let $\mathsf{Cat}$ be a category with finite product and terminal object $\star$. A group object in $\mathsf{Cat}$ is an object $G$ in $\mathsf{Cat}$ together with morphisms $$\mu :G\times G\to G,\eta :G\to G,ϵ:\star \to G$$such that the following diagrams commute

1.	associativity:
2.	unit:
3.	inverse: $\mu$ is called the multiplication, $\eta$ is called the inverse, $ϵ$ is called the unit.

Example 8.5. Here are some classical examples.

•	Group objects in $\mathsf{Set}$ are groups.
•	Group objects in $\mathsf{Top}$ are topological groups.
•	Group objects in $h\mathsf{Top}$ are called $\mathsf{H}$-groups.

Proposition 8.6. Let $\mathsf{Cat}$ be a category with finite products and a terminal object. Let $G$ be a group object. Then $$\mathrm{Hom}(-,G):\mathsf{Cat}\to \mathsf{Group}$$defines a contravariant functor from $\mathsf{Cat}$ to $\mathsf{Group}$.

Proof. For any $X\in \mathsf{Cat}$, we define the group structure on $\mathrm{Hom}(X,G)$ as followings:

•	Multiplication: $f\cdot g=\mu (f,g)$ as $$\begin{array}{rlrl}\mathrm{Hom}(X,G)& \times \mathrm{Hom}(X,G)& ⟶& \mathrm{Hom}(X,G)\\ (X\stackrel{f}{\to }G& ,X\stackrel{g}{\to }G)& ⟼& X\stackrel{(f,g)}{\to }G\times G\stackrel{\mu }{\to }G\end{array}$$
•	Inverse: $f^{-1}=\eta (f)$ as $$\begin{array}{rlr}\mathrm{Hom}(X,G)& ⟶& \mathrm{Hom}(X,G)\\ X\stackrel{f}{\to }G& ⟼& X\stackrel{f}{\to }G\stackrel{\eta }{\to }G,\end{array}$$
•	Identity is the image of the morphism $\mathrm{Hom}(X,\star )\to \mathrm{Hom}(X,G)$.

$\square$

Remark. The converse is also true, by Yoneda Lemma.

In the category $\underline{{\mathcal{T}}_{\star }}$, product exists and is given by$$(X,x_0)\times (Y,y_0)=(X\times Y,(x_0,y_0)).$$It admits a zero (both initial and terminal) object $\star$, which is a single point space.

Lemma 8.7. The quotient functor $\underline{{\mathcal{T}}_{\star }}\to \underline{h{\mathcal{T}}_{\star }}$ preserves finite product.

Proof. Exercise.

$\square$

Theorem 8.8. Let $X\in \underline{{\mathcal{T}}_{\star }}$. Then $\Omega X$ is a group object in $\underline{h{\mathcal{T}}_{\star }}$.

Proof. The multiplication is the $\star$ composition of paths as Definition 2.5 $$\Omega X\times \Omega X\to \Omega X.$$The inverse is the usual reverse of paths. The constant path $1_{x_0}$ is the zero object. The associativity follows from Proposition . We leave the detail to the readers.

$\square$

By Proposition 8.8, an immediate consequence is:

Corollary 8.9. Any $X\in \underline{h{\mathcal{T}}_{\star }}$ defines a functor $$[-,\Omega X{]}_0:\underline{h{\mathcal{T}}_{\star }}\to \mathsf{Group}.$$

Definition 8.10. Let $(X,x_0)\in \underline{{\mathcal{T}}_{\star }}$. We define its suspension $\Sigma X$ by the quotient of $X\times I$: The suspension is the same as the smash product with $S^1$ $$\Sigma X=S^1\wedge X.$$It defines functors $$\Sigma :\underline{{\mathcal{T}}_{\star }}\to \underline{{\mathcal{T}}_{\star }},\underline{h{\mathcal{T}}_{\star }}\to \underline{h{\mathcal{T}}_{\star }}.$$

Example 8.11. By Example 5.29, $\Sigma S^n\cong S^{n+1}$ are homeomorphic for any $n\ge 0$.

Theorem 8.12. $(\Sigma ,\Omega )$ defines adjoint pairs $$\Sigma :\underline{{\mathcal{T}}_{\star }}\rightleftarrows \underline{{\mathcal{T}}_{\star }}:\Omega \Sigma :\underline{h{\mathcal{T}}_{\star }}\rightleftarrows \underline{h{\mathcal{T}}_{\star }}:\Omega$$

Proof. It follows from Theorem 5.34.

$\square$

Definition 8.13. Let $(X,x_0)\in \underline{{\mathcal{T}}_{\star }}$. We define the $n$-th homotopy group $${\pi }_n(X,x_0):=[S^n,X{]}_0.$$Sometimes we simply denote it by ${\pi }_n(X)$.

In particular, we have

•	${\pi }_0$ is the path connected component.
•	${\pi }_1$ is the fundamental group.
•	For $n\ge 1$, we know that. $${\pi }_n(X)=[\Sigma S^{n-1},X{]}_0=[S^{n-1},\Omega X{]}_0$$which is a group since $\Omega X$ is a group object.

Proposition 8.14. ${\pi }_n(X)$ is abelian if $n\ge 2$.

Proof. This statement can be also illustrated as follows: The following statements are the analogue of what we did in Section 2.

$\square$

Proposition 8.15. Let $X$ be path connected. There is a natural functor $$T_n:{\Pi }_1(X)\to \mathsf{Group}$$which sends $x_0$ to ${\pi }_n(X,x_0)$. In particular, there is a natural action of ${\pi }_1(X,x_0)$ on ${\pi }_n(X,x_0)$ and all ${\pi }_n(X,x_0)$’s are isomorphic for different choices of $x_0$.

Proposition 8.16. Let $f:X\to Y$ be a homotopy equivalence. Then $$f_{\ast }:{\pi }_n(X,x_0)\to {\pi }_n(Y,f(x_0))$$is a group isomorphism.