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2

$begingroup$

I have a solid torus $T$ and a curve (or knot) $C$ that winds 2-times around the torus (parallel to the longitude). Can I say that $[C] = 2 in mathbbZ cong pi_1(T)$, that is the curve represents 2 in the fundamental group? Can I take the fundamental group without a basepoint, since I want for $C$ do not necessarily go through the basepoint?
In a paper I've heard the expression a "conjugacy class of $pi_1(T)$", if C' represents an element in this conjugacy class, how is it different than $C$?

algebraic-topology

share|cite|improve this question

asked Mar 13 at 0:48

Jake B.Jake B.

1786

$endgroup$

migrated from mathoverflow.net Mar 13 at 9:00

This question came from our site for professional mathematicians.

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  The expression in the paper was probably referring to the conjugacy class of an element in $pi_1 T$. Regarding the difference between the conjugacy classes and just elements in the fundamental group there are two issues: how change-of-basepoints affects fundamental group, and also the Hurewicz theorem about the map $pi_1 X to H_1 X$.
  $endgroup$
  – Ryan Budney
  Mar 13 at 1:55
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Depending on the choice of isomorphism $pi_1(T)cong mathbbZ$ your curve could also represent $-2$.
  $endgroup$
  – Greg Friedman
  Mar 13 at 5:02
  

  
  
  
  

add a comment | 

2

$begingroup$

I have a solid torus $T$ and a curve (or knot) $C$ that winds 2-times around the torus (parallel to the longitude). Can I say that $[C] = 2 in mathbbZ cong pi_1(T)$, that is the curve represents 2 in the fundamental group? Can I take the fundamental group without a basepoint, since I want for $C$ do not necessarily go through the basepoint?
In a paper I've heard the expression a "conjugacy class of $pi_1(T)$", if C' represents an element in this conjugacy class, how is it different than $C$?

algebraic-topology

share|cite|improve this question

asked Mar 13 at 0:48

Jake B.Jake B.

1786

$endgroup$

migrated from mathoverflow.net Mar 13 at 9:00

This question came from our site for professional mathematicians.

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  The expression in the paper was probably referring to the conjugacy class of an element in $pi_1 T$. Regarding the difference between the conjugacy classes and just elements in the fundamental group there are two issues: how change-of-basepoints affects fundamental group, and also the Hurewicz theorem about the map $pi_1 X to H_1 X$.
  $endgroup$
  – Ryan Budney
  Mar 13 at 1:55
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Depending on the choice of isomorphism $pi_1(T)cong mathbbZ$ your curve could also represent $-2$.
  $endgroup$
  – Greg Friedman
  Mar 13 at 5:02
  

  
  
  
  

add a comment | 

$begingroup$

I have a solid torus $T$ and a curve (or knot) $C$ that winds 2-times around the torus (parallel to the longitude). Can I say that $[C] = 2 in mathbbZ cong pi_1(T)$, that is the curve represents 2 in the fundamental group? Can I take the fundamental group without a basepoint, since I want for $C$ do not necessarily go through the basepoint?
In a paper I've heard the expression a "conjugacy class of $pi_1(T)$", if C' represents an element in this conjugacy class, how is it different than $C$?

algebraic-topology

share|cite|improve this question

asked Mar 13 at 0:48

Jake B.Jake B.

1786

$endgroup$

share|cite|improve this question

asked Mar 13 at 0:48

Jake B.Jake B.

1786

share|cite|improve this question

asked Mar 13 at 0:48

Jake B.Jake B.

1786

asked Mar 13 at 0:48

Jake B.Jake B.

1786

asked Mar 13 at 0:48

Jake B.Jake B.

1786