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A positive answer to this question would imply a positive answer to the open problem on existence of a "coincidence point" of two commuting maps $f_1, f_2: T\to T$, where $T$ is the triod (tripod), see Question 1 in McDowell's survey http://topology.auburn.edu/tp/reprints/v34/tp34025p1.pdf

The point is that if $f_1, f_2: T\to T$ are commuting maps as above, one can define maps $\tilde{f_i}=f_i\circ R: D^2\to D^2$ where $R: D^2\to T$ is a retraction. (I am assuming that $T$ is embedded in $D^2$.) Then $\tilde{f_1}, \tilde{f_2}$ commute if and only if $f_1$ and $f_2$ commute, furthermore, $f_1$ and $f_2$ have a coincidence point $x, f_1(x)=f_2(x),$ if and only if $\tilde{f_1}, \tilde{f_2}$ do.

Of course, if one were to look for counter-examples, it would be easier to find ones among maps of the 2-disk.

Incidentally, it appears that coincidence problem (in the general setting of compact manifolds) was first addressed by Lefschetz, see http://en.wikipedia.org/wiki/Lefschetz_fixed-point_theorem . Lefschetz formula for coincidence was extended to compact manifolds with boundary in 1980s, see references in Saveliev's paper http://arxiv.org/pdf/math.AT/9909028.pdf

A positive answer to this question would imply a positive answer to the open problem on existence of a "coincidence point" of two commuting maps $f_1, f_2: T\to T$, where $T$ is the triod (tripod), see Question 1 in McDowell's survey http://topology.auburn.edu/tp/reprints/v34/tp34025p1.pdf

The point is that if $f_1, f_2: T\to T$ are commuting maps as above, one can define maps $\tilde{f_i}=f_i\circ R: D^2\to D^2$ where $R: D^2\to T$ is a retraction. (I am assuming that $T$ is embedded in $D^2$.) Then $\tilde{f_1}, \tilde{f_2}$ commute if and only if $f_1$ and $f_2$ commute, furthermore, $f_1$ and $f_2$ have a coincidence point $x, f_1(x)=f_2(x),$ if and only if $\tilde{f_1}, \tilde{f_2}$ do.

Of course, if one were to look for counter-examples, it would be easier to find ones among maps of the 2-disk.

Incidentally, it appears that coincidence problem (in the general setting of compact manifolds) was first addressed by Lefschetz, see http://en.wikipedia.org/wiki/Lefschetz_fixed-point_theorem . Lefschetz formula for coincidence was extended to compact manifolds with boundary in 1980s, see references in Saveliev's paper http://arxiv.org/pdf/math.AT/9909028.pdf

A positive answer to this question would imply a positive answer to the open problem on existence of a "coincidence point" of two commuting maps $f_1, f_2: T\to T$, where $T$ is the triod (tripod), see Question 1 in McDowell's survey http://topology.auburn.edu/tp/reprints/v34/tp34025p1.pdf

The point is that if $f_1, f_2: T\to T$ are commuting maps as above, one can define maps $\tilde{f_i}=f_i\circ R: D^2\to D^2$ where $R: D^2\to T$ is a retraction. (I am assuming that $T$ is embedded in $D^2$.) Then $\tilde{f_1}, \tilde{f_2}$ commute if and only if $f_1$ and $f_2$ commute, furthermore, $f_1$ and $f_2$ have a coincidence point $x, f_1(x)=f_2(x),$ if and only if $\tilde{f_1}, \tilde{f_2}$ do.

Of course, if one were to look for counter-examples, it would be easier to find ones among maps of the 2-disk.

Incidentally, it appears that coincidence problem (in the general setting of compact manifolds) was first addressed by Lefschetz, see http://en.wikipedia.org/wiki/Lefschetz_fixed-point_theorem . Lefschetz formula for coincidence was extended to compact manifolds with boundary in 1980s, see references in Saveliev's paper http://arxiv.org/pdf/math.AT/9909028.pdf
   

A positive answer to this question would imply a positive answer to the open problem on existence of a "coincidence point" of two commuting maps $f_1, f_2: T\to T$, where $T$ is the triod (tripod), see Question 1 in McDowell's survey http://topology.auburn.edu/tp/reprints/v34/tp34025p1.pdf

The point is that if $f_1, f_2: T\to T$ are commuting maps as above, one can define maps $\tilde{f_i}=f_i\circ R: D^2\to D^2$ where $R: D^2\to T$ is a retraction. (I am assuming that $T$ is embedded in $D^2$.) Then $\tilde{f_1}, \tilde{f_2}$ commute if and only if $f_1$ and $f_2$ commute, furthermore, $f_1$ and $f_2$ have a coincidence point $x, f_1(x)=f_2(x),$ if and only if $\tilde{f_1}, \tilde{f_2}$ do.

Of course, if one were to look for counter-examples, it would be easier to find ones among maps of the 2-disk.

Incidentally, it appears that coincidence problem (in the general setting of compact manifolds) was first addressed by Lefschetz, see http://en.wikipedia.org/wiki/Lefschetz_fixed-point_theorem . Lefschetz formula for coincidence was extended to compact manifolds with boundary in 1980s, see references in Saveliev's paper http://arxiv.org/pdf/math.AT/9909028.pdf

A positive answer to this question would imply a positive answer to the open problem on existence of a "coincidence point" of two commuting maps $f_1, f_2: T\to T$, where $T$ is the triod (tripod), see Question 1 in McDowell's survey http://topology.auburn.edu/tp/reprints/v34/tp34025p1.pdf

The point is that if $f_1, f_2: T\to T$ are commuting maps as above, one can define maps $\tilde{f_i}=f_i\circ R: D^2\to D^2$ where $R: D^2\to T$ is a retraction. (I am assuming that $T$ is embedded in $D^2$.) Then $\tilde{f_1}, \tilde{f_2}$ commute if and only if $f_1$ and $f_2$ commute, furthermore, $f_1$ and $f_2$ have a coincidence point $x, f_1(x)=f_2(x),$ if and only if $\tilde{f_1}, \tilde{f_2}$ do.

Of course, if one were to look for counter-examples, it would be easier to find ones among maps of the 2-disk.

Incidentally, it appears that coincidence problem (in the general setting of compact manifolds) was first addressed by Lefschetz, see http://en.wikipedia.org/wiki/Lefschetz_fixed-point_theorem . Lefschetz formula for coincidence was extended to compact manifolds with boundary in 1980s, see references in Saveliev's paper http://arxiv.org/pdf/math.AT/9909028.pdf

A positive answer to this question would imply a positive answer to the open problem on existence of a "coincidence point" of two commuting maps $f_1, f_2: T\to T$, where $T$ is the triod (tripod), see Question 1 in McDowell's survey http://topology.auburn.edu/tp/reprints/v34/tp34025p1.pdf

The point is that if $f_1, f_2: T\to T$ are commuting maps as above, one can define maps $\tilde{f_i}=f_i\circ R: D^2\to D^2$ where $R: D^2\to T$ is a retraction. (I am assuming that $T$ is embedded in $D^2$.) Then $\tilde{f_1}, \tilde{f_2}$ commute if and only if $f_1$ and $f_2$ commute, furthermore, $f_1$ and $f_2$ have a coincidence point $x, f_1(x)=f_2(x),$ if and only if $\tilde{f_1}, \tilde{f_2}$ do.

Of course, if one were to look for counter-examples, it would be easier to find ones among maps of the 2-disk.

Incidentally, it appears that coincidence problem (in the general setting of compact manifolds) was first addressed by Lefschetz, see http://en.wikipedia.org/wiki/Lefschetz_fixed-point_theorem . Lefschetz formula for coincidence was extended to compact manifolds with boundary in 1980s, see references in Saveliev's paper http://arxiv.org/pdf/math.AT/9909028.pdf