0708-1300/Class notes for Thursday, January 24

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Typed Notes

The notes below are by the students and for the students. Hopefully they are useful, but they come with no guarantee of any kind.

Proof of Van Kampen

Let $G_{i}=\pi _{1}(U_{i})$

$H=\pi _{1}(Y_{1}\cap U_{2})$

$G=\pi _{1}(U_{1}\cup U_{2})$

We aim to show that $G=G_{1}*_{H}G_{2}$

Hence, we want to define two maps:

$\Phi :G_{1}*_{H}G_{2}\rightarrow G$ and

$\Psi :G\rightarrow G_{1}*_{H}G_{2}$

such that they are inverses of each other.

Now, recall the commuting diagram:

${\begin{matrix}&\ \ \ \ U_{1}&&\\&\nearrow ^{i_{1}}&\searrow ^{j_{1}}&\\U_{1}\cap U_{2}&&&U_{1}\cup U_{2}=X\\&\searrow _{i_{2}}&\nearrow ^{j_{2}}&\\&\ \ \ \ U_{2}&&\\\end{matrix}}$

Further, let $b_{i}$ alternate between 1 and 2 for successive i's.

Hence we define $\Phi$ via for $\alpha _{i}\in G_{b_{i}}$,

[$\alpha _{1}][\alpha _{2}]\ldots [\alpha _{n}]\rightarrow [j_{b_{1}*}\alpha _{1}\cdot \ldots \cdot j_{b_{n}*}\alpha _{n}]$

Clearly this is well defined. We need to check the relations in $G_{1}*_{H}G_{2}$ indeed hold. Well, the identity element corresponds to the identity path so the relation that removes identities holds. Furthermore, the concatenation of paths is a sum after $\Phi$ and the definition necessitates the third relation holds.

Now for $\Psi$:

Elements in G correspond with paths $\gamma$ in $U_{1}\cup U_{2}$

On consider such a $\gamma$. The Lesbegue Lemma let us break $\gamma$ up so $\gamma =\gamma _{1}\ldots \gamma _{N}$ such that each $\gamma _{i}$ is in just $G_{b_{i}}$

Now, $\gamma _{i}$ does not go from base point to base point, so we can't consider it as a loop itself. Let $x_{i}$ denote the endpoint of $\gamma _{i}$ (and hence $x_{i-1}$ is the beginning point) and further, $x_{0}=b$ the base point.

Choose paths $\eta _{i}$ connecting $x_{i}$ to b such that if $x_{i}\in U_{b_{i}}$ then $\eta _{i}\subset U_{b_{i}}$

Hence, $\gamma \sim {\bar {\eta _{0}}}\gamma _{1}\eta _{1}{\bar {\eta _{1}}}\gamma _{2}\eta _{2}\ldots \mapsto [{\bar {\eta _{0}}}\gamma _{1}\eta _{1}][{\bar {\eta _{1}}}\gamma _{2}\eta _{2}]\ldots$

where each section is entirely in $G_{b_{i}}$. This above mapping is $\Psi$

We need to show that $\Psi$ is well defined. I.e.,

1) $\Psi$ is independent of the subdivision

2) $\Psi$ is independent of choice of the Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \eta_i} 's

3) If Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \gamma_1\sim\gamma_2} then Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Psi(\gamma_1) = \Psi(\gamma_2)}

For 1) it is enough to show that one can add or remove a single subdivision point. Suppose you add a new subdivision point c in between two others a and b. Traveling between a and b is the same as going from a to c then to the basepoint and back and then continuing on to b. But this is precisely what happens when you add a subdivision point. Likewise for removing points.

For 2), Suppose Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \eta_i} goes from the basepoint to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x_i} . Add a subdivision point y right beside Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x_i} and remove the one at Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x_i} . Then add again the original point Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x_i} . The trick is that the new Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \eta_i'} is the one used when adding y. The map from the new $x_{i}$ to the basepoint is this Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \eta_i'} with the infinitesimal connection between y and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x_i} added. The subdivision point y is then removed leaving our original configuration of basepoints, only with the path from Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x_i} to b now being Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \eta_i'} instead of the original Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \eta_i}

For 3) we consider the homotopy between Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \gamma_1} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \gamma_2} thought of as a square with Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \gamma_1} on the bottom and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \gamma_2} on the top. We use the Lesbegue Lemma to subdivide the square into many subsquares such that each one is entirely in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle U_{b_i}}

We further modify the homotopy H to a new homotopy Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \tilde{H}} such that each grid point gets mapped to b.

This can be thought of as "pinching" each gird point and pulling it to b or, alternatively, as tossing a "handkerchief on a bed of nails"

If we let Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \gamma_1} be broken in to section $\alpha _{1},\ldots ,\alpha _{n}$ along the bottom then Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \tilde{H}} first lifts Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \alpha_1} to the top and right sides of the subsquare such that the gridpoint goes to b. Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \tilde{H}} then moves the next square up in an analogous manner until we are at the top with Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \gamma_2} .