- #1

Euge

Gold Member

MHB

POTW Director

- 1,993

- 1,346

Prove that if there is a fiber bundle ##S^k \to S^m \to S^n##, then ##k = n-1## and ##m = 2n-1##.

You are using an out of date browser. It may not display this or other websites correctly.

You should upgrade or use an alternative browser.

You should upgrade or use an alternative browser.

- POTW
- Thread starter Euge
- Start date

- #1

Euge

Gold Member

MHB

POTW Director

- 1,993

- 1,346

Prove that if there is a fiber bundle ##S^k \to S^m \to S^n##, then ##k = n-1## and ##m = 2n-1##.

- #2

Infrared

Science Advisor

Gold Member

- 996

- 556

We may assume that ##n>0## since for ##S^m\to S^n## to be surjective, the codomain ##S^n## needs to be connected, and it's not possible to have ##m=0.##

Case 1: ##n>1.## We examine the following part of the associated long exact sequence:

##\pi_{k+1}(S^m)\to\pi_{k+1}(S^n)\to\pi_k(S^k)\to\pi_k(S^m).## The outer groups are zero since ##\pi_i(S^j)=0## when ##0<i<j.## So, ##\pi_{k+1}(S^n)\cong\mathbb{Z}## and hence ##k+1\geq n.##

We now consider the following part of the same exact sequence:

##\pi_n(S^m)\to\pi_n(S^n)\to\pi_{n-1}(S^k)\to\pi_{n-1}(S^m).## The case ##n=m## is impossible because that would make ##k=0,## in which case ##S^m\to S^m## is a double cover, but ##n=m>1## so ##S^n## is simply connected. So, again the two outer groups vanish and ##\pi_{n-1}(S^k)\cong\mathbb{Z},## giving ##n-1\geq k.## Together with the above, this means ##k=n-1.##

Case 2: ##n=1.## Examine the tail end of the exact sequence: ##\pi_1(S^m)\to\pi_1(S^1)\to\pi_0(S^k)## (where ##\pi_0## isn't given a group structure but exactness still makes sense and is valid.) If ##m>1## then the image of ##\pi_1(S^m)\to\pi_1(S^1)## is trivial and cannot match the kernel of ##\pi_1(S^1)\to\pi_0(S^k)##, which is the whole ##\pi_1(S^1).## So ##m=1,## which forces ##k=0## and the triple ##(k,m,n)=(0,1,1)## satisfies the right equations.

It's been a while since I took algebraic topology but hopefully this is mostly right.

Last edited by a moderator:

Share:

- Last Post

- Replies
- 5

- Views
- 414

- Last Post

- Replies
- 1

- Views
- 266

- Last Post

- Replies
- 2

- Views
- 118

- Last Post

- Replies
- 6

- Views
- 513

- Last Post

- Replies
- 1

- Views
- 257

- Last Post

- Replies
- 1

- Views
- 286

- Last Post

- Replies
- 3

- Views
- 221

- Last Post

- Replies
- 0

- Views
- 15

- Last Post

- Replies
- 4

- Views
- 430

- Last Post

- Replies
- 1

- Views
- 714