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In this math overflow question asked almost 5 years ago about the (normalized) Lusternik-Schnirelmann category of $4$-manifolds, the second-last comment (by Jeff Strom) says the following:

A $4$-dimensional compact manifold with free $\pi_1$ has L-S category $2$.

Specifically, if $\pi_1(M) = 0$ and $M \ne S^4$ is a $4$-manifold, then the above is true (as remarked by Mark Grant in the above question post). But I am not sure why the above statement is true for any $4$-manifold $M$ with $\pi_1(M)$ free and non-trivial. As observed by Oprea-Strom, it can be deduced from the work of Matumoto-Katanaga (and Hellen Colman) that the LS category of such manifolds is at most $2$. I don't understand why their LS category should be exactly $2$ (in particular, why can't it be $1$). Can someone please provide a reason/reference?

In this math overflow question asked almost 5 years ago about the (normalized) Lusternik-Schnirelmann category of $4$-manifolds, the second-last comment (by Jeff Strom) says the following:

A $4$-dimensional compact manifold with free $\pi_1$ has L-S category $2$.

Specifically, if $\pi_1(M) = 0$ and $M \ne S^4$ is a $4$-manifold, then the above is true (as remarked by Mark Grant in the above question post). But I am not sure why the above statement is true for any $4$-manifold $M$ with $\pi_1(M)$ free and non-trivial. As observed by Oprea–Strom, it can be deduced from the work of Matumoto–Katanaga (and Hellen Colman) that the LS category of such manifolds is at most $2$. I don't understand why their LS category should be exactly $2$ (in particular, why can't it be $1$). Can someone please provide a reason/reference?

In this math overflow question asked almost 5 years ago about the Lusternik-Schnirelmann category of $4$-manifolds, the second-last comment (by Jeff Strom) says the following:

A $4$-dimensional compact manifold with free $\pi_1$ has L-S category $2$.

Specifically, if $\pi_1(M) = 0$ and $M \ne S^4$ is a $4$-manifold, then the above is true (as remarked by Mark Grant in the above question post). But I am not sure why the above statement is true for any $4$-manifold $M$ with $\pi_1(M)$ free and non-trivial. As observed by Oprea-Strom, it can be deduced from the work of Matumoto-Katanaga (and Hellen Colman) that the LS category of such manifolds is at most $2$. I don't understand why their LS category should be exactly $2$. Can someone please provide a reason/reference?

In this math overflow question asked almost 5 years ago about the (normalized) Lusternik-Schnirelmann category of $4$-manifolds, the second-last comment (by Jeff Strom) says the following:

A $4$-dimensional compact manifold with free $\pi_1$ has L-S category $2$.

Specifically, if $\pi_1(M) = 0$ and $M \ne S^4$ is a $4$-manifold, then the above is true (as remarked by Mark Grant in the above question post). But I am not sure why the above statement is true for any $4$-manifold $M$ with $\pi_1(M)$ free and non-trivial. As observed by Oprea-Strom, it can be deduced from the work of Matumoto-Katanaga (and Hellen Colman) that the LS category of such manifolds is at most $2$. I don't understand why their LS category should be exactly $2$ (in particular, why can't it be $1$). Can someone please provide a reason/reference?

In this math overflow question asked almost 5 years ago about the Lusternik-Schnirelmann category of $4$-manifolds, the second-last comment (by Jeff Strom) says the following:

A $4$-dimensional compact manifold with free $\pi_1$ has L-S category $2$.

Specifically, if $\pi_1(M) = 0$ and $M \ne S^4$ is a $4$-manifold, then the above is true (as remarked by Mark Grant in the above question post). But I am not sure why the above statement is true for any $4$-manifold $M$ with $\pi_1(M)$ free and non-trivial. As observed by Oprea-Strom, it can be deduced from the work of Matumoto-Katanaga (and Collman) that the LS category of such manifolds is at most $2$. I don't understand why their LS category should be exactly $2$. Can someone please provide a reason/reference?

In this math overflow question asked almost 5 years ago about the Lusternik-Schnirelmann category of $4$-manifolds, the second-last comment (by Jeff Strom) says the following:

A $4$-dimensional compact manifold with free $\pi_1$ has L-S category $2$.

Specifically, if $\pi_1(M) = 0$ and $M \ne S^4$ is a $4$-manifold, then the above is true (as remarked by Mark Grant in the above question post). But I am not sure why the above statement is true for any $4$-manifold $M$ with $\pi_1(M)$ free and non-trivial. As observed by Oprea-Strom, it can be deduced from the work of Matumoto-Katanaga (and Hellen Colman) that the LS category of such manifolds is at most $2$. I don't understand why their LS category should be exactly $2$. Can someone please provide a reason/reference?