an almost complex structure on $M$, compatible with the orientation, if and only if it is an integral lift of the second Stiefel–Whitney class and $$\label{eq:hirzebruch}
c^2=2\chi+3\sigma.$$ (The necessity of equation  is the Hirzebruch signature theorem.) Given such a class $c$, denote by $${{\mathcal J}}(M,c) := \left\{J\in{{\mathrm{Aut}}}(TM)\,\bigg|\,
\begin{array}{l}
J^2=-{{{\mathchoice {\rm 1\mskip-4mu l} {\rm 1\mskip-4mu l}
{\rm 1\mskip-4.5mu l} {\rm 1\mskip-5mu l}}}},\,c_1(TM,J)=c,\\
J\mbox{ induces the given orientation}
\end{array}
\right\}$$ the space of almost complex structures on $M$ with first Chern class $c$ and compatible with the orientation of $M$. If $M$ is simply connected then ${{\mathcal J}}(M,c)$ has precisely two connected components for every $c\in{{\mathcal C}}$. In general, there is a bijection $$\label{eq:mrowka}
\pi_0({{\mathcal J}}(M,c))\cong 
\mathrm{Tor}_2(H^2(M;{{\mathbb Z}}))\times
\left({{\mathbb Z}}/2{{\mathbb Z}}\oplus\frac{H^3(M;{{\mathbb Z}})}{H^1(M;{{\mathbb Z}})\cup c}\right)$$ for every $c\in{{\mathcal C}}$. (This was pointed out to me a long time ago by Tom Mrowka, but I don’t know a reference.) The first factor is the set $$\mathrm{Tor}_2(H^2(M;{{\mathbb Z}})):=\left\{a\in H^2(M;{{\mathbb Z}})\,|\,2a=0\right\}.$$ It characterizes the isomorphism classes of spin$^c$ structures on $TM$ with first Chern class $c$ (see \[para:SW\] and Lawson–Michelsohn [@LM Appendix D]). The second factor characterizes the set of homotopy classes of almost complex structures on $M$ whose canonical spin$^c$ structures are isomorphic to a 