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I have a (basic?) question in topology.

Question 1. Is it possible to characterise compact $4$-manifolds $M^4$, such that almost complex structures on $M^4$ are uniquely defined up to homotopy by their first Chern classes? Or is there at least a large class of $4$-manifolds where this is true? Maybe there is a reference?

Recall that if we denote by $\tau$ the signature of $M^4$ and by $e$ its Euler characteristics then for any element $c\in H^2(M^4,\mathbb Z)$ with $c^2=3\tau+2e$ there is an almost complex structure $J$ on $M^4$, such that $c_1(M,J)=c$. However this $J$ might be non-unique, up to homotopy among almost complex structures with $c_1=c$, as the following example shows.

Example. Let $M^4=S^1\times S^3$, then the tangent bundle is trivial, so almost complex structures on $M^4$ can be identified up to homotopy with homotopy classes of maps $M^4\to S^2$ (see page 11 in The geometry of Four-manifolds Donaldson Kronheimer). Maps $S^3\to S^2$ can have different Hopf invariants, but $c_1=0$ since $S^1\times S^3$ has no second cohomology...

Added Question 2. Is there at least one manifold that satisfies condition of Question 1?

In McDuff-Salamon (footnote on page 120) it is written, that if $M^4$ is spin then there are precisely two homotopy classes of $J$ with given $c_1$ (this is said to be related to $\pi_4(S^2)=\mathbb Z_2$). But since $S^1\times S^3$ is spin this statement from McDuff-Salamon seem to contradict to my conclusion (that for $S^1\times S^3$ homotopy classes are can have different Hopf invariants). So, where is the mistake?...:)

I have a (basic?) question in topology.

Question 1. Is it possible to characterise compact $4$-manifolds $M^4$, such that almost complex structures on $M^4$ are uniquely defined up to homotopy by their first Chern classes? Or is there at least a large class of $4$-manifolds where this is true? Maybe there is a reference?

Recall the theorem of Wu. Denote by $\tau$ the signature of $M^4$ and by $e$ the Euler characteristics. Then for any element $c\in H^2(M^4,\mathbb Z)$ with $c^2=3\tau+2e$ and such that $c\; {\rm mod} \;\mathbb Z_2= w_2\in H^2(M^4,\mathbb Z_2)$ there is an almost complex structure $J$ on $M^4$, such that $c_1(M,J)=c$ (in particular, as Paul says $e+\tau=0\; {\rm mod} \;4$). However this $J$ might be non-unique, up to homotopy among almost complex structures with $c_1=c$, as the following example shows.

Example. Let $M^4=S^1\times S^3$, then the tangent bundle is trivial, so almost complex structures on $M^4$ can be identified up to homotopy with homotopy classes of maps $M^4\to S^2$ (see page 11 in The geometry of Four-manifolds Donaldson Kronheimer). Maps $S^3\to S^2$ can have different Hopf invariants, but $c_1=0$ since $S^1\times S^3$ has no second cohomology...

Added Question 2. Is there at least one manifold that satisfies condition of Question 1?

In McDuff-Salamon (footnote on page 120) it is written, that if $M^4$ is spin then there are precisely two homotopy classes of $J$ with given $c_1$ (this is said to be related to $\pi_4(S^2)=\mathbb Z_2$). But since $S^1\times S^3$ is spin this statement from McDuff-Salamon seem to contradict to my conclusion (that for $S^1\times S^3$ homotopy classes are can have different Hopf invariants). So, where is the mistake?...:)

I have a (basic?) question in topology.

Question 1. Is it possible to characterise compact $4$-manifolds $M^4$, such that almost complex structures on $M^4$ are uniquely defined up to homotopy by their first Chern classes? Or is there at least a large class of $4$-manifolds where this is true? Maybe there is a reference?

Recall that if we denote by $\tau$ the signature of $M^4$ and by $e$ its Euler characteristics then for any element $c\in H^2(M^4,\mathbb Z)$ with $c^2=3\tau+2e$ there is an almost complex structure $J$ on $M^4$, such that $c_1(M,J)=c$. However this $J$ might be non-unique, up to homotopy among almost complex structures with $c_1=c$, as the following example shows.

Example. Let $M^4=S^1\times S^3$, then the tangent bundle is trivial, so almost complex structures on $M^4$ can be identified up to homotopy with homotopy classes of maps $M^4\to S^2$ (see page 11 in The geometry of Four-manifolds Donaldson Kronheimer). Maps $S^3\to S^2$ can have different Hopf invariants, but $c_1=0$ since $S^1\times S^3$ has no second cohomology...

Added Question 2. Is there at least one manifold that satisfies condition of Question 1?

In Mcduff-Salamon (footnote on page 120) it is written, that if $M^4$ is spin then there are precisely two homotopy classes of $J$ with given $c_1$ (this is said to be related to $\pi_4(S^2)=\mathbb Z_2$). But since $S^1\times S^3$ is spin this statement from McDuff-Salamon seem to contradict to my conclusion (that for $S^1\times S^3$ homotopy classes are characterized by Hopf invariant). So, where is the mistake?...:)

I have a (basic?) question in topology.

Question 1. Is it possible to characterise compact $4$-manifolds $M^4$, such that almost complex structures on $M^4$ are uniquely defined up to homotopy by their first Chern classes? Or is there at least a large class of $4$-manifolds where this is true? Maybe there is a reference?

Recall that if we denote by $\tau$ the signature of $M^4$ and by $e$ its Euler characteristics then for any element $c\in H^2(M^4,\mathbb Z)$ with $c^2=3\tau+2e$ there is an almost complex structure $J$ on $M^4$, such that $c_1(M,J)=c$. However this $J$ might be non-unique, up to homotopy among almost complex structures with $c_1=c$, as the following example shows.

Example. Let $M^4=S^1\times S^3$, then the tangent bundle is trivial, so almost complex structures on $M^4$ can be identified up to homotopy with homotopy classes of maps $M^4\to S^2$ (see page 11 in The geometry of Four-manifolds Donaldson Kronheimer). Maps $S^3\to S^2$ can have different Hopf invariants, but $c_1=0$ since $S^1\times S^3$ has no second cohomology...

Added Question 2. Is there at least one manifold that satisfies condition of Question 1?

In McDuff-Salamon (footnote on page 120) it is written, that if $M^4$ is spin then there are precisely two homotopy classes of $J$ with given $c_1$ (this is said to be related to $\pi_4(S^2)=\mathbb Z_2$). But since $S^1\times S^3$ is spin this statement from McDuff-Salamon seem to contradict to my conclusion (that for $S^1\times S^3$ homotopy classes are can have different Hopf invariants). So, where is the mistake?...:)

I have a (basic?) question in topology.

Question 1. Is it possible to characterise compact $4$-manifolds $M^4$, such that almost complex structures on $M^4$ are uniquely defined up to homotopy by their first Chern classes? Or is there at least a large class of $4$-manifolds where this is true? Maybe there is a reference?

Recall that if we denote by $\tau$ the signature of $M^4$ and by $e$ its Euler characteristics then for any element $c\in H^2(M^4,\mathbb Z)$ with $c^2=3\tau+2e$ there is an almost complex structure $J$ on $M^4$, such that $c_1(M,J)=c$. However this $J$ might be non-unique, up to homotopy among almost complex structures with $c_1=c$, as the following example shows.

Example. Let $M^4=S^1\times S^3$, then the tangent bundle is trivial, so almost complex structures on $M^4$ can be identified up to homotopy with homotopy classes of maps $M^4\to S^2$ (see page 11 in The geometry of Four-manifolds Donaldson Kronheimer). Maps $S^3\to S^2$ can have different Hopf invariants, but $c_1=0$ since $S^1\times S^3$ has no second cohomology...

I have a (basic?) question in topology.

Question 1. Is it possible to characterise compact $4$-manifolds $M^4$, such that almost complex structures on $M^4$ are uniquely defined up to homotopy by their first Chern classes? Or is there at least a large class of $4$-manifolds where this is true? Maybe there is a reference?

Recall that if we denote by $\tau$ the signature of $M^4$ and by $e$ its Euler characteristics then for any element $c\in H^2(M^4,\mathbb Z)$ with $c^2=3\tau+2e$ there is an almost complex structure $J$ on $M^4$, such that $c_1(M,J)=c$. However this $J$ might be non-unique, up to homotopy among almost complex structures with $c_1=c$, as the following example shows.

Example. Let $M^4=S^1\times S^3$, then the tangent bundle is trivial, so almost complex structures on $M^4$ can be identified up to homotopy with homotopy classes of maps $M^4\to S^2$ (see page 11 in The geometry of Four-manifolds Donaldson Kronheimer). Maps $S^3\to S^2$ can have different Hopf invariants, but $c_1=0$ since $S^1\times S^3$ has no second cohomology...

Added Question 2. Is there at least one manifold that satisfies condition of Question 1?

In Mcduff-Salamon (footnote on page 120) it is written, that if $M^4$ is spin then there are precisely two homotopy classes of $J$ with given $c_1$ (this is said to be related to $\pi_4(S^2)=\mathbb Z_2$). But since $S^1\times S^3$ is spin this statement from McDuff-Salamon seem to contradict to my conclusion (that for $S^1\times S^3$ homotopy classes are characterized by Hopf invariant). So, where is the mistake?...:)