My main question in this post here is that:
How can we relate the two following indices mod 2:

$ eta $ invariant,

the number of zero modes of the Dirac operator,
associated with the StiefelWhitney class $ w_2 (V_ {SO (3)}) $ of the associated vector package of $ SO (3) $,
regarding local data like the 2nd class of Chern $ c_2 $ of principal $ US (2) $ package, or the socalled $ F_a wedge F_a $ term of $ US (2) $ field strength $ F_a $?
$$
exp big ({i pi} int limits_ {M_4} c_2 big)
=
exp big ( frac {i pi} {8 pi ^ 2} int limits_ {M_4} text {Tr} , F_a wedge F_a big)
$$
(Note that by local datawe mean for example locally well defined data, eg. field strength / curvature measured by performing local parallel transport of the main packet SU (2).)
Detailed context of the questions:
Follow Annals of Physics 394, 244293 (2018), DOI: 10.1016 / j.aop.2018.04.025, Guo, Putrov and Wang, I know that

the $ eta $ invariant I referred to above is from one $ mathbb Z_4 $ class
in 4d ofa a bordering group defined as
$$ Omega _ { mathrm {Pin} ^ {+} times _ { mathbb Z_2} SU (2)} ^ 4 equiv text {Hom} ( Omega ^ { mathrm {Pin} ^ {+ } times _ { mathbb Z_2} SU (2)} _ 4, U (1)) Z_4 times mathbb Z_2 $$
$$
Z ^ { nu}[a]= exp (2 pi i nu eta_ {SU (2)}[a])
qquad ( nu) in mathbb Z_4.
$$ 
The number of zero modes of the above mentioned Dirac operator comes from a $ mathbb Z_2 $ class
in 4d's
$$ Omega _ { mathrm {Pin} ^ {} times _ { mathbb Z_2} SU (2)} ^ 4 equiv text {Hom} ( Omega ^ { mathrm {Pin} ^ { } times _ { mathbb Z_2} SU (2)} _ 4, U (1)) cong ( mathbb Z_2) ^ 3. $$
$$
Z ^ { nu} = ( 1) ^ { nu N_0}, qquad ( nu) in mathbb Z_2.
$$ More details: in this case, each eigenvalue of the Dirac operator is accompanied by an opposite value (ie by presenting an operator who does not commute with the l & # 39; 39; operator Dirac and who commutes with the transition functions). A similar calculation then gives:
begin {equation}
Z ^ { nu} _ {SU (2)}[a] = left ( frac { det ({D} _a  m )} { det ({D} _a +  m )} right) ^ nu
stackrel { m  rightarrow infty} { longrightarrow} (1) ^ { nu N_0 & # 39;
end {equation}
or $ N_0 $ is the number of zero modes of the operator Dirac. Its mod 2 value is a topological spin invariant known as the mod 2 index. Nontrivial fSPT classes generated by such Dirac fermions are effectively labeled by $ nu in mathbb Z_2 $.
 More details: in this case, each eigenvalue of the Dirac operator is accompanied by an opposite value (ie by presenting an operator who does not commute with the l & # 39; 39; operator Dirac and who commutes with the transition functions). A similar calculation then gives:
The notations of the bordism / cobordism group are quite classic, where we also follow the above reference Annals of Physics 394, 244293 (2018) and references therein.