Dgdrxrt

How do Java 8 default methods hеlp with lambdas?

How could a hydrazine and N2O4 cloud (or it's reactants) show up in weather radar?

What is a more techy Technical Writer job title that isn't cutesy or confusing?

Why complex landing gears are used instead of simple, reliable and light weight muscle wire or shape memory alloys?

Do i imagine the linear (straight line) homotopy in a correct way?

New Order #6: Easter Egg

draw a pulley system

Can I cut the hair of a conjured korred with a blade made of precious material to harvest that material from the korred?

First paper to introduce the "principal-agent problem"

Pointing to problems without suggesting solutions

NIntegrate on a solution of a matrix ODE

Can gravitational waves pass through a black hole?

As a dual citizen, my US passport will expire one day after traveling to the US. Will this work?

Where and when has Thucydides been studied?

One-one communication

Keep at all times, the minus sign above aligned with minus sign below

3D Masyu - A Die

Does a random sequence of vectors span a Hilbert space?

How to ask rejected full-time candidates to apply to teach individual courses?

How can I list files in reverse time order by a command and pass them as arguments to another command?

Noise in Eigenvalues plot

How does TikZ render an arc?

Why can't fire hurt Daenerys but it did to Jon Snow in season 1?

Shimano 105 brifters (5800) and Avid BB5 compatibility

Two maps from $S^1$ to $S^1$ are not homotopic

1

$begingroup$

I'm reading the book Algebraic Topology from a Homotopical Viewpoint but I don't understand why the $ n in mathbbZ $ in Proposition 2.4.4 is unique. It suffices to prove that $ widehatphi_n $ and $ widehatphi_m $ are not homotopic where $$ widehatphi_k : mathbbS^1 rightarrow mathbbS^1 qquad widehatphi_n (e^i2pi t) = e^i2kpi t $$ for $ n neq m $. Is this obvious ?

algebraic-topology

share|cite|improve this question

edited Apr 2 at 16:25

Andrews

1,3062423

asked Apr 2 at 15:05

Math FanaticMath Fanatic

444

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  If they were homotopic their lifts would end at the same spot. They don't.
  $endgroup$
  – Randall
  Apr 2 at 15:06
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  @Randall But in this book it does not define lift before this proposition.
  $endgroup$
  – Math Fanatic
  Apr 2 at 15:10
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  View a (pointed) homotopy $F:S^1times Irightarrow S^1$, $hatvarphi_nsimeqhatvarphi_m$, by adjunction as a continuous family of maps $Ini tmapsto F_t:S^1rightarrow S^1$. Apply the propositions to get a continuous family of maps $varphi_t:Irightarrow mathbbR$. Then $tmapstovarphi_t(1)$ is a path in $mathbbR$ from $varphi_n(1)=n$ to $varphi_m(1)=m$. However, since $F$ is pointed $hatvarphi_t(1)=1in S^1$ for all $t$. Hence $varphi_t(1)inmathbbZ$ for all $t$. Hence $varphi_t(1)=n$ for all $t$. Hence $m=n$.
  $endgroup$
  – Tyrone
  Apr 2 at 16:15
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  @Tyrone How to prove that $ t mapsto phi_t(1) $ is continuous ? and why the homotopy $ F $ of $ widehatphi_n $ and $ widehatphi_m $ must satisfy $ F(1, t) = 1 forall tin I $ ?
  $endgroup$
  – Math Fanatic
  Apr 2 at 22:46
  
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Continuity follows from the exponential law (section 1.3). It's not true that a homtopy $F$ has to satisfy $F(1,t)=1$ for all $t$, but using the methods of Theorem 2.4.2 you can show that you can always choose $F$ to satisfy this condition.
  $endgroup$
  – Tyrone
  Apr 3 at 8:36
  
  

  
  
  
  

add a comment | 

1

$begingroup$

I'm reading the book Algebraic Topology from a Homotopical Viewpoint but I don't understand why the $ n in mathbbZ $ in Proposition 2.4.4 is unique. It suffices to prove that $ widehatphi_n $ and $ widehatphi_m $ are not homotopic where $$ widehatphi_k : mathbbS^1 rightarrow mathbbS^1 qquad widehatphi_n (e^i2pi t) = e^i2kpi t $$ for $ n neq m $. Is this obvious ?

algebraic-topology

share|cite|improve this question

edited Apr 2 at 16:25

Andrews

1,3062423

asked Apr 2 at 15:05

Math FanaticMath Fanatic

444

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  If they were homotopic their lifts would end at the same spot. They don't.
  $endgroup$
  – Randall
  Apr 2 at 15:06
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  @Randall But in this book it does not define lift before this proposition.
  $endgroup$
  – Math Fanatic
  Apr 2 at 15:10
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  View a (pointed) homotopy $F:S^1times Irightarrow S^1$, $hatvarphi_nsimeqhatvarphi_m$, by adjunction as a continuous family of maps $Ini tmapsto F_t:S^1rightarrow S^1$. Apply the propositions to get a continuous family of maps $varphi_t:Irightarrow mathbbR$. Then $tmapstovarphi_t(1)$ is a path in $mathbbR$ from $varphi_n(1)=n$ to $varphi_m(1)=m$. However, since $F$ is pointed $hatvarphi_t(1)=1in S^1$ for all $t$. Hence $varphi_t(1)inmathbbZ$ for all $t$. Hence $varphi_t(1)=n$ for all $t$. Hence $m=n$.
  $endgroup$
  – Tyrone
  Apr 2 at 16:15
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  @Tyrone How to prove that $ t mapsto phi_t(1) $ is continuous ? and why the homotopy $ F $ of $ widehatphi_n $ and $ widehatphi_m $ must satisfy $ F(1, t) = 1 forall tin I $ ?
  $endgroup$
  – Math Fanatic
  Apr 2 at 22:46
  
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Continuity follows from the exponential law (section 1.3). It's not true that a homtopy $F$ has to satisfy $F(1,t)=1$ for all $t$, but using the methods of Theorem 2.4.2 you can show that you can always choose $F$ to satisfy this condition.
  $endgroup$
  – Tyrone
  Apr 3 at 8:36
  
  

  
  
  
  

add a comment | 

$begingroup$

I'm reading the book Algebraic Topology from a Homotopical Viewpoint but I don't understand why the $ n in mathbbZ $ in Proposition 2.4.4 is unique. It suffices to prove that $ widehatphi_n $ and $ widehatphi_m $ are not homotopic where $$ widehatphi_k : mathbbS^1 rightarrow mathbbS^1 qquad widehatphi_n (e^i2pi t) = e^i2kpi t $$ for $ n neq m $. Is this obvious ?

algebraic-topology

share|cite|improve this question

edited Apr 2 at 16:25

Andrews

1,3062423

asked Apr 2 at 15:05

Math FanaticMath Fanatic

444

$endgroup$

share|cite|improve this question

edited Apr 2 at 16:25

Andrews

1,3062423

asked Apr 2 at 15:05

Math FanaticMath Fanatic

444

share|cite|improve this question

edited Apr 2 at 16:25

Andrews

1,3062423

asked Apr 2 at 15:05

Math FanaticMath Fanatic

444

edited Apr 2 at 16:25

Andrews

1,3062423

edited Apr 2 at 16:25

Andrews

1,3062423

asked Apr 2 at 15:05

Math FanaticMath Fanatic

444

asked Apr 2 at 15:05

Math FanaticMath Fanatic

444