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Hatcher's Exercise 1.2.8 is the following : Compute the fundamental group of the space $X$ obtained from two tori $S^1 \times S^1$ by identifying a circle $S^1 \times \{x_0\}$ in one torus with the corresponding circle $S^1 \times \{x_0\}$ in the other torus.

There are solutions on this site already, as in the discussion here, where the original poster mentions that one needs open sets in order to apply the van Kampen theorem. I'd like to follow up on this thought with the solution I found here (and also shown below).

enter image description here

Here are my questions about the provided solution:

  • What allows us to take an open neighborhood of $S^1 \times \{x_0\}$ that deformation retracts onto $S^1 \times \{x_0\}$? Does any space admit such an open neighborhood that deformation retracts onto it? Or is there something special about $S^1 \times \{x_0\}$ that admits this open neighborhood?
  • Applying the van Kampen theorem here requires $X$ to be the union of two path-connected open sets with path-connected intersection. How is $X$ being written as a union of two path-connected open sets here? Is $X = (T_1 \cup U_1) \cup (T_2 \cup U_2)$? If so, how can we see that $T_1 \cup U_1$ and $T_2 \cup U_2$ are path-connected and open, and have path-connected intersection?
  • Below is the statement of the van Kampen theorem in Hatcher. With this in mind, it looks like the normal subgroup $N$ in the provided solution is $\pi_1(S^1)$. How can one see that this is the case? enter image description here

Thanks!

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  • $\begingroup$ You can embed $X$ into $\Bbb R^3$ as a smaller torus inside the hole of a larger torus, with them tangent along the inner circle of the larger torus and outer circle of the inner torus. This common circle acts as $S^1 \times \{x_0\}$. $\endgroup$ May 24 '21 at 16:14
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  • In the more general case of two copies $A \times B$ joined by identifying the two copies of $A \times \{x_0\}$ for some $x_0 \in B$, as long as $x_0$ has a neighborhood $U$ in $B$ that deformation retracts to $\{x_0\}$, then $A \times U$ is a neighborhood of $A \times \{x_0\}$ that deformation retracts to $A \times \{x_0\}$. Every point of $S^1$ has neighborhoods isomophic to intervals on the real line. Intervals are contractible.
  • There is a slip-up on the indexing in this part of the proof. Since $U_1 \subset T_1, T_1 \cup U_1 = T_1$ and similarly $T_2 \cup U_2 = T_2$. What they meant was the two open sets are $T_1 \cup U_2, T_2 \cup U_1$. Let $C = T_1 \cap T_2$ be the common circle. $T_1 \cup U_2$ is open since it is the image under the quotient of the open set $T_1 \sqcup U_2$ in $T_1 \sqcup T_2$. Similarly for $T_2\cup U_1$. Since $U_2$ retracts to $C$, $T_1 \cup U_2$ retracts to $T_1 \cup C = T_1$ and $U_1 \cup U_2$ retracts to $U_1 \cup C = U_1$, which itself retracts to $C$. Similarly $T_2 \cup U_1$ retracts to $T_2$. Since $C \cong S^1$, it is path-connected, and $T_1, T_2$ are path-connected. Since $U_1 \cup U_2$ retracts to $C$, it is also path-connected.
  • The retractions above induce group isomorphisms $$\pi_1(T_1 \cup U_2) \cong \pi_1(T_1)\\\pi_1(T_2 \cup U_1) \cong \pi_1(T_2)\\\pi_1(U_1 \cup U_2) \cong \pi_1(C) \cong \pi_1(S^1)$$
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  • $\begingroup$ Excellent. One last question -- how do we know that the normal subgroup $N$ here is $\pi_1(U_1 \cup U_2)$? $\endgroup$ May 24 '21 at 18:19
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    $\begingroup$ Let $x_0$ be on $C$, and let $C_1, C_2 = S^1 \times \{x_0\}$ in $T_1, T_2$ respectively. $T_1 \cong C_1 \times C, T_2 \cong C_2 \times C$ and $\pi_1(T_1) \cong \pi_1(C_1) \times \pi_1(C)$ - that is ,every loop in $T_1$ at $x_0$ is equivalent to going around $C_1$ a few times, then going around $C$ a few times. Similarly for $T_2$. But it is the same circle $C$ for both. Trips around $C$ commute with trips around either $C_1$ or $C_2$. So in $\pi_1(T_1) * \pi_1(T_2)$, you can collect all the trips around $C$ together, leaving onlty $\pi_1(C_1) * \pi_1(C_2)$. $\endgroup$ May 25 '21 at 1:44
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    $\begingroup$ I admit that isn't a well-stated argument. My algebra skills are pretty weak anymore, but my geometric intuition has gotten stronger, so I tend to think in that direction. $\endgroup$ May 25 '21 at 1:45

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