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Concrete example Example of closed 4 manifold with $\mathbb{S}^1$ action with 1 fixed point and free away from it

I am looking for a smooth closed 4-manifold $M$ with a distinguished point $x$$x\in M$, endowed with an $\mathbb{S}^1$ action such that the stabilizer of $p\in M\setminus\{x\}$ is trivial and $x$ is fixed.

A naive attempt:

If we consider the action given by $\mathbb{S}^1$ on $\mathbb{S}^3\subset \mathbb{C}^2$ that gives the Hopf fibration we can extend this action to the 4-ball (it is an unitary action), and it will have the origin as single fixed point. Unfortunately this manifold is not closed.

Concrete example of closed 4 manifold with $\mathbb{S}^1$ action with 1 fixed point and free away from it

I am looking for a smooth closed 4-manifold $M$ with a distinguished point $x$, endowed with an $\mathbb{S}^1$ action such that the stabilizer of $p\in M\setminus\{x\}$ is trivial and $x$ is fixed.

A naive attempt:

If we consider the action given by $\mathbb{S}^1$ on $\mathbb{S}^3\subset \mathbb{C}^2$ that gives the Hopf fibration we can extend this action to the 4-ball (it is an unitary action), and it will have the origin as single fixed point. Unfortunately this manifold is not closed.

Example of closed 4 manifold with $\mathbb{S}^1$ action with 1 fixed point and free away from it

I am looking for a smooth closed 4-manifold $M$ with a distinguished point $x\in M$, endowed with an $\mathbb{S}^1$ action such that the stabilizer of $p\in M\setminus\{x\}$ is trivial and $x$ is fixed.

A naive attempt:

If we consider the action given by $\mathbb{S}^1$ on $\mathbb{S}^3\subset \mathbb{C}^2$ that gives the Hopf fibration we can extend this action to the 4-ball (it is an unitary action), and it will have the origin as single fixed point. Unfortunately this manifold is not closed.

Source Link
Overflowian
Overflowian
  • 2.5k
  • 1
  • 12
  • 25

Concrete example of closed 4 manifold with $\mathbb{S}^1$ action with 1 fixed point and free away from it

I am looking for a smooth closed 4-manifold $M$ with a distinguished point $x$, endowed with an $\mathbb{S}^1$ action such that the stabilizer of $p\in M\setminus\{x\}$ is trivial and $x$ is fixed.

A naive attempt:

If we consider the action given by $\mathbb{S}^1$ on $\mathbb{S}^3\subset \mathbb{C}^2$ that gives the Hopf fibration we can extend this action to the 4-ball (it is an unitary action), and it will have the origin as single fixed point. Unfortunately this manifold is not closed.