Is there exists any non-trivial space (i.e not deformation retract onto a point) in $\mathbb R^n$ such that any continuous map from the space onto itself has a fixed point. I highly suspect that quasi circle on $\mathbb R^2$ is an example. Yet I've not written down the (dirty) proof. But in this case its all homotpy groups are trivial. So if I assume my space as a manifold, then (QESTION:) is this fixed point property forces it to become a contractible manifold. I read some where that there exists a contractible compact manifold which does not satisfy this fixed point property. So is there exists any non-contractible manifold (compact) where this property follows? Or otherwise can anyone please provide an outline of how to prove that such a manifold is contractible?

Does there exist any non-trivial space (i.e not deformation retract onto a point) in $\mathbb R^n$ such that any continuous map from the space onto itself has a fixed point. I highly suspect that the quasi circle on $\mathbb R^2$ is an example. Yet I've not written down the (dirty) proof. But in this case all its homotpy groups are trivial. So if I assume my space as a manifold, then (QUESTION:) does this fixed point property force it to become a contractible manifold? I read somewhere that there exists a contractible compact manifold which does not satisfy this fixed point property. So does there exist any non-contractible manifold (compact) where this property follows? Or otherwise can anyone please provide an outline of how to prove that such a manifold is contractible?

Is there exists any non-trivial space (i.e not deformation retract onto a point) in $\mathbb R^n$ such that any continuous map from the space onto itself has a fixed point. I highly suspect that quasi circle on $\mathbb R^2$ is an example. Yet I've not written down the (dirty) proof. But in this case its all homotpy groups are trivial. So if I assume my space as a manifold, then (QESTION:) is this fixed point property force it to become a contractible manifold. I read some where that there exists a contractible compact manifold which does not satisfy this fixed point property. So is there exists any non-contractible manifold (compact) where this property follows? Or otherwise can anyone please provide an outline of how to prove that such a manifold is contractible?

Is there exists any non-trivial space (i.e not deformation retract onto a point) in $\mathbb R^n$ such that any continuous map from the space onto itself has a fixed point. I highly suspect that quasi circle on $\mathbb R^2$ is an example. Yet I've not written down the (dirty) proof. But in this case its all homotpy groups are trivial. So if I assume my space as a manifold, then (QESTION:) is this fixed point property forces it to become a contractible manifold. I read some where that there exists a contractible compact manifold which does not satisfy this fixed point property. So is there exists any non-contractible manifold (compact) where this property follows? Or otherwise can anyone please provide an outline of how to prove that such a manifold is contractible?