1
$\begingroup$

This question from Hatcher's Algebraic topology was discussed and answered in this post. However, both the original asker's method and the correct answer's method seem believable. Why is one correct and the other isn't? (they arrive at different conclusions namely $F_{n}$ and $F_{2n-1}$) Furthermore, is there any way to reconcile these two approaches so that they both arrive at the correct conclusion?

Thanks in advance!

$\endgroup$

1 Answer 1

Reset to default
2
$\begingroup$

A line passing through the cylinder punctures it in two places, not one. So for every new line added, you add two wedges of circles (up to homotopy), not one as the OP suggests. Induction gives the rest.

$\endgroup$
2
  • $\begingroup$ I see, but shouldn't that make the answer 2n not 2n-1? $\endgroup$
    – elevensor
    Mar 3, 2022 at 2:43
  • $\begingroup$ The first line only creates a single circle because puncturing a sphere once doesn't change the fundamental group; it just 'opens it up' into a disk. You need the second puncture to then turn the disk into a circle (up to homotopy). This is why base cases are so important in induction! $\endgroup$
    – Dan Rust
    Mar 3, 2022 at 11:03

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .