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Some comments.

1. The space of sections of $LX \to X$ is a grouplike topological monoid, by pointwise multiplication. So, its path components do form a group. In fact it is a loop space $\Omega_1F(X,X)$ of the space of functions $F(X,X)$, where loops are based at the identity map $1\!\!\! : X \to X$.

2. For $A\to X$ a map, let $\Gamma(LX|A)$ denote the space of sections of $LX \to X$ along $A$. This is the same as the space of sections of the pullback $A \times_X LX \to A$. This is also a loop space, this time of $F(A,X)$.

When $A = X$ we set $\Gamma(LX) = \Gamma(LX|A)$.

3. The restriction map $$ F(X,X) \to F(A,X) $$ is a fibration and it loops to the restriction map $\Gamma(LX) \to \Gamma(LX|A)$ . Its fiber at the constant map $A\to X$ is the based function space $F_*(X/A,X)$.

4. Let's apply the above to the inclusion $\ast \subset S^2$ basepoint. This gives a homotopy fiber sequence $$ \Omega^2 S^2 \to F(S^2,S^2) \to S^2\, , $$ where the second map is given by evaluation.

5. Take $\pi_*$ (with respect to the correct basepoint "1") to get an exact sequence of groups $$ \cdots \to \pi_2(S^2) \to \pi_1(\Omega^2S^2;1) \to \pi_1(F(S^2,S^2);1) \to 1 $$ where $\pi_1(F(S^2,S^2);1) = \pi_0(LS^2)$, and $\pi_2(S^2) = \Bbb Z = \pi_1(\Omega^2S^2;1)$. It follows that there's a short exact sequence of groups $$ \Bbb Z \to \Bbb Z \to \pi_0(LS^2) \to 1\, . $$ It therefore suffices to compute the homomorphism $\Bbb Z \to \Bbb Z$. To compute this, you'll need to understand how the boundary operator works. This amounts in the end to the computation discussed by "user95545" above.

Some comments.

1. The space of sections of $LX \to X$ is a grouplike topological monoid, by pointwise multiplication. So, its path components do form a group. In fact it is a loop space $\Omega_1F(X,X)$ of the space of functions $F(X,X)$, where loops are based at the identity map $1\!\!\! : X \to X$.

2. For $A\to X$ a map, let $\Gamma(LX|A)$ denote the space of sections of $LX \to X$ along $A$. This is the same as the space of sections of the pullback $A \times_X LX \to A$. This is also a loop space, this time of $F(A,X)$.

When $A = X$ we set $\Gamma(LX) = \Gamma(LX|A)$.

3. The restriction map $$ F(X,X) \to F(A,X) $$ is a fibration and it loops to the restriction map $\Gamma(LX) \to \Gamma(LX|A)$ . Its fiber at the constant map $A\to X$ is the based function space $F_*(X/A,X)$.

4. Let's apply the above to the inclusion $\ast \subset S^2$ basepoint. This gives a homotopy fiber sequence $$ \Omega^2 S^2 \to F(S^2,S^2) \to S^2\, , $$ where the second map is given by evaluation.

5. Take $\pi_*$ (with respect to the correct basepoint "1") to get an exact sequence of groups $$ \cdots \to \pi_2(S^2) \to \pi_1(\Omega^2S^2;1) \to \pi_1(F(S^2,S^2);1) \to 1 $$ where $\pi_1(F(S^2,S^2);1) = \pi_0(\Gamma(LS^2))$, and $\pi_2(S^2) = \Bbb Z = \pi_1(\Omega^2S^2;1)$. It follows that there's a short exact sequence of groups $$ \Bbb Z \to \Bbb Z \to \pi_0(\Gamma(LS^2)) \to 1\, . $$ It therefore suffices to compute the homomorphism $\Bbb Z \to \Bbb Z$. To compute this, you'll need to understand how the boundary operator works. This amounts in the end to the computation discussed by "user95545" above.

Some comments.

1. The space of sections of $LX \to X$ is a grouplike topological monoid, by pointwise multiplication. So, its path components do form a group. In fact it is a loop space $\Omega_1F(X,X)$ of the space of functions $F(X,X)$, where loops are based at the identity map $1\!\!\! : X \to X$.

2. For $A\to X$ a map, let $\Gamma(LX|A)$ denote the space of sections of $LX \to X$ along $A$. This is the same as the space of sections of the pullback $A \times_X LX \to A$. This is also a loop space, this time of $F(A,X)$.

When $A = X$ we set $\Gamma(LX) = \Gamma(LX|A)$.

3. The restriction map $$ F(X,X) \to F(A,X) $$ is a fibration and it loops to the restriction map $\Gamma(LX) \to \Gamma(LX|A)$ . Its fiber at the constant map $A\to X$ is the based function space $F_*(X/A,X)$.

4. Let's apply the above to the inclusion $\ast \subset S^2$ basepoint. This gives a homotopy fiber sequence $$ \Omega^2 S^2 \to F(S^2,S^2) \to S^2\, , $$ where the second map is given by evaluation.

5. Take $\pi_*$ (with respect to the correct basepoint "1") to get an exact sequence of groups $$ \cdots \to \pi_2(S^2) \to \pi_1(\Omega^2S^2;1) \to \pi_1(F(S^2,S^2);1) \to 1 $$ where $\pi_1(F(S^2,S^2);1) = \pi_0(LS^2)$, and $\pi_2(S^2) = \Bbb Z = \pi_1(\Omega^2S^2;1)$. It follows that there's a short exact sequence of groups $$ \Bbb Z \to \Bbb Z \to \pi_0(LS^2) \to 1\, . $$ It therefore suffices to compute the homomorphism $\Bbb Z \to \Bbb Z$. To compute this, you'll need to understand how the boundary operator works.

Some comments.

1. The space of sections of $LX \to X$ is a grouplike topological monoid, by pointwise multiplication. So, its path components do form a group. In fact it is a loop space $\Omega_1F(X,X)$ of the space of functions $F(X,X)$, where loops are based at the identity map $1\!\!\! : X \to X$.

2. For $A\to X$ a map, let $\Gamma(LX|A)$ denote the space of sections of $LX \to X$ along $A$. This is the same as the space of sections of the pullback $A \times_X LX \to A$. This is also a loop space, this time of $F(A,X)$.

When $A = X$ we set $\Gamma(LX) = \Gamma(LX|A)$.

3. The restriction map $$ F(X,X) \to F(A,X) $$ is a fibration and it loops to the restriction map $\Gamma(LX) \to \Gamma(LX|A)$ . Its fiber at the constant map $A\to X$ is the based function space $F_*(X/A,X)$.

4. Let's apply the above to the inclusion $\ast \subset S^2$ basepoint. This gives a homotopy fiber sequence $$ \Omega^2 S^2 \to F(S^2,S^2) \to S^2\, , $$ where the second map is given by evaluation.

5. Take $\pi_*$ (with respect to the correct basepoint "1") to get an exact sequence of groups $$ \cdots \to \pi_2(S^2) \to \pi_1(\Omega^2S^2;1) \to \pi_1(F(S^2,S^2);1) \to 1 $$ where $\pi_1(F(S^2,S^2);1) = \pi_0(LS^2)$, and $\pi_2(S^2) = \Bbb Z = \pi_1(\Omega^2S^2;1)$. It follows that there's a short exact sequence of groups $$ \Bbb Z \to \Bbb Z \to \pi_0(LS^2) \to 1\, . $$ It therefore suffices to compute the homomorphism $\Bbb Z \to \Bbb Z$. To compute this, you'll need to understand how the boundary operator works. This amounts in the end to the computation discussed by "user95545" above.

Some comments.

1. The space of sections of $LX \to X$ is a grouplike topological monoid, by pointwise multiplication. So, its path components do form a group. In fact it is a loop space $\Omega_1F(X,X)$ of the space of functions $F(X,X)$, where loops are based at the identity map $1\!\!\! : X \to X$.

2. For $A\to X$ a map, let $\Gamma(LX|A)$ denote the space of sections of $LX \to X$ along $A$. This is the same as the space of sections of the pullback $A \times_X LX \to A$. This is also a loop space, this time of $F(A,X)$.

When $A = X$ we set $\Gamma(LX) = \Gamma(LX|A)$.

3. The restriction map $$ F(X,X) \to F(A,X) $$ is a fibration and it loops to the restriction map $\Gamma(LX) \to \Gamma(LX|A)$ . Its fiber at the constant map $A\to X$ is the based function space $F_*(X/A,X)$.

4. Let's apply the above to the inclusion $D_+^2 \subset S^2$ given by the northern hemisphere. This gives a homotopy fiber sequence $$ \Omega^2 S^2 \to F(S^2,S^2) \to F(D^2_+,S^2) \simeq S^2\, . $$

5. Take $\pi_*$ (with respect to the correct basepoint "1") to get an exact sequence of groups $$ \cdots \to \pi_2(S^2) \to \pi_1(\Omega^2S^2;1) \to \pi_1(F(S^2,S^2);1) \to 1 $$ where $\pi_1(F(S^2,S^2);1) = \pi_0(LS^2)$, and $\pi_2(S^2) = \Bbb Z = \pi_1(\Omega^2S^2;1)$. It follows that there's a short exact sequence of groups $$ \Bbb Z \to \Bbb Z \to \pi_0(LS^2) \to 1\, . $$ It therefore suffices to compute the homomorphism $\Bbb Z \to \Bbb Z$. To compute this, you'll need to understand how the boundary operator works.

Some comments.

1. The space of sections of $LX \to X$ is a grouplike topological monoid, by pointwise multiplication. So, its path components do form a group. In fact it is a loop space $\Omega_1F(X,X)$ of the space of functions $F(X,X)$, where loops are based at the identity map $1\!\!\! : X \to X$.

2. For $A\to X$ a map, let $\Gamma(LX|A)$ denote the space of sections of $LX \to X$ along $A$. This is the same as the space of sections of the pullback $A \times_X LX \to A$. This is also a loop space, this time of $F(A,X)$.

When $A = X$ we set $\Gamma(LX) = \Gamma(LX|A)$.

3. The restriction map $$ F(X,X) \to F(A,X) $$ is a fibration and it loops to the restriction map $\Gamma(LX) \to \Gamma(LX|A)$ . Its fiber at the constant map $A\to X$ is the based function space $F_*(X/A,X)$.

4. Let's apply the above to the inclusion $\ast \subset S^2$ basepoint. This gives a homotopy fiber sequence $$ \Omega^2 S^2 \to F(S^2,S^2) \to S^2\, , $$ where the second map is given by evaluation.

5. Take $\pi_*$ (with respect to the correct basepoint "1") to get an exact sequence of groups $$ \cdots \to \pi_2(S^2) \to \pi_1(\Omega^2S^2;1) \to \pi_1(F(S^2,S^2);1) \to 1 $$ where $\pi_1(F(S^2,S^2);1) = \pi_0(LS^2)$, and $\pi_2(S^2) = \Bbb Z = \pi_1(\Omega^2S^2;1)$. It follows that there's a short exact sequence of groups $$ \Bbb Z \to \Bbb Z \to \pi_0(LS^2) \to 1\, . $$ It therefore suffices to compute the homomorphism $\Bbb Z \to \Bbb Z$. To compute this, you'll need to understand how the boundary operator works.