Skip to main content
MathOverflow

Return to Question

fix
Source Link
Ilya Nikokoshev
Ilya Nikokoshev
  • 15.1k
  • 12
  • 77
  • 129

Origin

This question was asked by John Baez in This Week's Finds in Mathematical Physics (Week 286). Therefore, please don't upvote this question (unless you really want to), but do upvote the answers.

Background/Motivation

For a CW complex, which (here for simplicity willwe'll have $\pi_1 = 0$), you can do the operation of "rationalizing", which will change its homotopy $\pi_n \to \pi_n \otimes \mathbb Q$. This works by attaching enough cylinders so that each original cell is killed, but its subdivisions are born instead.

Question

Does there exist a similar procedure of "killing the torsion" which would change the homotopy of 1-connected CW complex from $\pi_n$ to $\pi_n/\pi_n^{tors}$?

Thoughts

One encounters problems if one just tries to kill off the cell: the procedure might have changed higher homology (this doesn't happen in rationalizing since cylinders are simple). So I suspect the answer is "No", but how to construct a counterexample?

Origin

This question was asked by John Baez in This Week's Finds in Mathematical Physics (Week 286). Therefore, please don't upvote this question (unless you really want to), but do upvote the answers.

Background/Motivation

For a CW complex, which for simplicity will have $\pi_1 = 0$, you can do the operation of "rationalizing", which will change its homotopy $\pi_n \to \pi_n \otimes \mathbb Q$. This works by attaching enough cylinders so that each original cell is killed, but its subdivisions are born instead.

Question

Does there exist a similar procedure of "killing the torsion" which would change the homotopy of 1-connected CW complex from $\pi_n$ to $\pi_n/\pi_n^{tors}$?

Thoughts

One encounters problems if one just tries to kill off the cell: the procedure might have changed higher homology (this doesn't happen in rationalizing since cylinders are simple). So I suspect the answer is "No", but how to construct a counterexample?

Origin

This question was asked by John Baez in This Week's Finds in Mathematical Physics (Week 286). Therefore, please don't upvote this question (unless you really want to), but do upvote the answers.

Background/Motivation

For a CW complex (here for simplicity we'll have $\pi_1 = 0$), you can do the operation of "rationalizing", which will change its homotopy $\pi_n \to \pi_n \otimes \mathbb Q$. This works by attaching enough cylinders so that each original cell is killed, but its subdivisions are born instead.

Question

Does there exist a similar procedure of "killing the torsion" which would change the homotopy of 1-connected CW complex from $\pi_n$ to $\pi_n/\pi_n^{tors}$?

Thoughts

One encounters problems if one just tries to kill off the cell: the procedure might have changed higher homology (this doesn't happen in rationalizing since cylinders are simple). So I suspect the answer is "No", but how to construct a counterexample?

Source Link
Ilya Nikokoshev
Ilya Nikokoshev
  • 15.1k
  • 12
  • 77
  • 129

Killing the torsion in homotopy

Origin

This question was asked by John Baez in This Week's Finds in Mathematical Physics (Week 286). Therefore, please don't upvote this question (unless you really want to), but do upvote the answers.

Background/Motivation

For a CW complex, which for simplicity will have $\pi_1 = 0$, you can do the operation of "rationalizing", which will change its homotopy $\pi_n \to \pi_n \otimes \mathbb Q$. This works by attaching enough cylinders so that each original cell is killed, but its subdivisions are born instead.

Question

Does there exist a similar procedure of "killing the torsion" which would change the homotopy of 1-connected CW complex from $\pi_n$ to $\pi_n/\pi_n^{tors}$?

Thoughts

One encounters problems if one just tries to kill off the cell: the procedure might have changed higher homology (this doesn't happen in rationalizing since cylinders are simple). So I suspect the answer is "No", but how to construct a counterexample?