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Ali Taghavi
Ali Taghavi
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Regarding the last part of your question:

Let $f_{0}$ and $f_{1}$ be two orientation preserving homeomorphism in $H_{+}(S^{1})$. Lift $f_{0}$ and $f_{1}$ to homeomorphisms $F_{0}$ and $F_{1}$ on $\mathbb{R}$, as in http://en.wikipedia.org/wiki/Rotation_number.

Then consider the quotient of the homotopy $F_{t}=tF_{0}+(1-t)F_{1}$ as a path of homeomorphism on $S^{1}$.

More precisely let $P:\mathbb{R}\to S^{1}$ be the standard quotient map. Then $G_{t}=P \circ F_{t}$ is a path of maps from $\mathbb{R}$ to $S^{1}$, which satisfies $G_{t}(x+1)=G_{t}$. In the quotient space $\mathbb{R}/P=S^{1}$ we obtain a path of orientation preserving homeomorphisms which connects $f_{0}$ to $f_{1}$.

Let $f_{0}$ and $f_{1}$ be two orientation preserving homeomorphism in $H_{+}(S^{1})$. Lift $f_{0}$ and $f_{1}$ to homeomorphisms $F_{0}$ and $F_{1}$ on $\mathbb{R}$, as in http://en.wikipedia.org/wiki/Rotation_number.

Then consider the quotient of the homotopy $F_{t}=tF_{0}+(1-t)F_{1}$ as a path of homeomorphism on $S^{1}$.

More precisely let $P:\mathbb{R}\to S^{1}$ be the standard quotient map. Then $G_{t}=P \circ F_{t}$ is a path of maps from $\mathbb{R}$ to $S^{1}$, which satisfies $G_{t}(x+1)=G_{t}$. In the quotient space $\mathbb{R}/P=S^{1}$ we obtain a path of orientation preserving homeomorphisms which connects $f_{0}$ to $f_{1}$.

Regarding the last part of your question:

Let $f_{0}$ and $f_{1}$ be two orientation preserving homeomorphism in $H_{+}(S^{1})$. Lift $f_{0}$ and $f_{1}$ to homeomorphisms $F_{0}$ and $F_{1}$ on $\mathbb{R}$, as in http://en.wikipedia.org/wiki/Rotation_number.

Then consider the quotient of the homotopy $F_{t}=tF_{0}+(1-t)F_{1}$ as a path of homeomorphism on $S^{1}$.

More precisely let $P:\mathbb{R}\to S^{1}$ be the standard quotient map. Then $G_{t}=P \circ F_{t}$ is a path of maps from $\mathbb{R}$ to $S^{1}$, which satisfies $G_{t}(x+1)=G_{t}$. In the quotient space $\mathbb{R}/P=S^{1}$ we obtain a path of orientation preserving homeomorphisms which connects $f_{0}$ to $f_{1}$.

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Ali Taghavi
Ali Taghavi
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Let $f$$f_{0}$ and $g$$f_{1}$ be two orientionorientation preserving homeomorphism in $H_{+}(S^{1})$. Lift $f$$f_{0}$ and $g$$f_{1}$ to homeomorphisms $F$$F_{0}$ and $G$$F_{1}$ on $\mathbb{R}$, as in http://en.wikipedia.org/wiki/Rotation_number. Then

Then consider the quotient of $tF+(1-t)G$the homotopy $F_{t}=tF_{0}+(1-t)F_{1}$ as a path of homeomorphism on $S^{1}$.

More precisely let $P:\mathbb{R}\to S^{1}$ be the standard quotient map. Then $G_{t}=P \circ F_{t}$ is a path of maps from $\mathbb{R}$ to $S^{1}$, which satisfies $G_{t}(x+1)=G_{t}$. In the quotient space $\mathbb{R}/P=S^{1}$ we obtain a path of orientation preserving homeomorphisms which connects $f_{0}$ to $f_{1}$.

Let $f$ and $g$ be two oriention preserving homeomorphism in $H_{+}(S^{1})$. Lift $f$ and $g$ to homeomorphisms $F$ and $G$ on $\mathbb{R}$, as in http://en.wikipedia.org/wiki/Rotation_number. Then consider the quotient of $tF+(1-t)G$ as a path of homeomorphism on $S^{1}$.

Let $f_{0}$ and $f_{1}$ be two orientation preserving homeomorphism in $H_{+}(S^{1})$. Lift $f_{0}$ and $f_{1}$ to homeomorphisms $F_{0}$ and $F_{1}$ on $\mathbb{R}$, as in http://en.wikipedia.org/wiki/Rotation_number.

Then consider the quotient of the homotopy $F_{t}=tF_{0}+(1-t)F_{1}$ as a path of homeomorphism on $S^{1}$.

More precisely let $P:\mathbb{R}\to S^{1}$ be the standard quotient map. Then $G_{t}=P \circ F_{t}$ is a path of maps from $\mathbb{R}$ to $S^{1}$, which satisfies $G_{t}(x+1)=G_{t}$. In the quotient space $\mathbb{R}/P=S^{1}$ we obtain a path of orientation preserving homeomorphisms which connects $f_{0}$ to $f_{1}$.

Source Link
Ali Taghavi
Ali Taghavi
  • 356
  • 8
  • 31
  • 123

Let $f$ and $g$ be two oriention preserving homeomorphism in $H_{+}(S^{1})$. Lift $f$ and $g$ to homeomorphisms $F$ and $G$ on $\mathbb{R}$, as in http://en.wikipedia.org/wiki/Rotation_number. Then consider the quotient of $tF+(1-t)G$ as a path of homeomorphism on $S^{1}$.