Homotopy Groups

September 24, 2024
By Aathreya Kadambi
Expanding on Lectures by Professor Alexander Givental and Fomenko and Fuchs

$\pi_n(X,x_0) = \pi_b((S^n,0), (X,x_0))$. Note that $\pi_0(X,x_0)$ is the number of path-connected components of $X$.

He then explained the pictures from Fig 37 and Fig 38 in the book and basically section 8.1.

Depednence on Base Points

If we have another base point $x_1$ and a path from $x_0$ to $x_1$ $\gamma$, we get a map $\pi_n(X,x_0)\xrightarrow{\gamma} \pi_n(X,x_1) \xrightarrow{\tilde{\gamma}} \pi_n(X,x_0)$.

We introduced many algebraic invariants but we can’t compute many of them unless the space is trivial, so let’s look at a covering $(T,t_0) \rightarrow (X,x_0)$. In the theory of coverings we showed that mappings $p : (T,t_0)\rightarrow (X,x_0)$ are injective. This is because if a loop is contractible in $(X,x_0)$, we can use the homotopy lifting lemma to lift this homotopy up to $T$, and then because the fiber is discrete, we get the same point up there so that the other loop upstairs is also contractible. From this, we get that $p_* : \pi_n(T,t_0) \hookrightarrow \pi_n(X,x_0)$ is injective for $n \ge 1$. It turns out for $n \ge 2$, it is also surjective.

Relative Homotopy Groups

He discussed some stuff from this section.

The Reason This Was introduced

This isn’t necessarily helpful for computations or anything, but it let’s us make the long exact homotopy sequence.

He wrote something about $A \xhookrightarrow{i} X$ and $(X,x_0,x_0) \xrightarrow{j}(X,A,x_0)$.

$\pi_0(X)\xleftarrow{i_*}\pi_0(A)\xleftarrow{\partial} \pi_1(X,A) \xleftarrow{j_*}\pi_1(X) \xleftarrow{i_*}\pi_1(A) \leftarrow \pi_2(X,A)\leftarrow \dots$

$\dots \xleftarrow{i_*} \pi_{n-1}(A)\xleftarrow{\partial} \pi_n(X,A)\xleftarrow{j_*}\pi_n(X) \xleftarrow{i_*}\pi_n(A) \xleftarrow{\partial}\pi_{n+1}(X,A) \leftarrow$