In this note, we want to correct a mistake in the proof of [1, Theorem 2.2] which does not affect the statement of the theorem, though. In [1, (2.7)], we incorrectly state an isomorphism of complexes, thus we need to modify the corresponding argument, by first replacing the last six lines on p. 1098, beginning with “we also have \(\cdots \)”, with the following text.
“We also have a \(\mathbb {Z}_2\)-equivariant short exact sequence of complexes
$$\begin{aligned} 0\rightarrow C^{\bullet } (W^u/W^u_{Y_{1}},F) \otimes \mathbb {C}^- \mathop {\longrightarrow }\limits ^{\gamma } C^{\bullet } (\widetilde{W}^u, \widetilde{F}) \mathop {\longrightarrow }\limits ^{\gamma } C^{\bullet }(W^u, F) \otimes \mathbb {C}^+\rightarrow 0 , \end{aligned}$$
(2.7)
given by
$$\begin{aligned} \gamma (\mathfrak {b}^*\otimes 1_{\mathbb {C}^-})\! =\! \frac{\sqrt{2}}{2} \Big ((j_{1}^{-1}|_{X_1})^*\mathfrak {b}^* - (j_{2}^{-1}|_{X_2})^*\mathfrak {b}^*\Big ) ,\,\, \gamma ( \mathfrak {a}^*)\!=\! \frac{\sqrt{2}}{2} \Big (j_{1}^* \mathfrak {a}^*|_{X_1}+ j_{2}^* \mathfrak {a}^*|_{X_2}\Big )\! \otimes \! 1_{\mathbb {C}^+} , \end{aligned}$$
(2.8)
with \(X_i =j_i(X)\), which induces a \(\mathbb {Z}_{2}\)-equivariant short exact sequence
$$\begin{aligned} 0\rightarrow H^\bullet (X, Y_1, F) \otimes \mathbb {C}^- \mathop {\longrightarrow }\limits ^{\gamma } H^\bullet (\widetilde{X}, \widetilde{F})\mathop {\longrightarrow }\limits ^{\gamma } H^\bullet (X,F) \otimes \mathbb {C}^+\rightarrow 0.\text {''} \end{aligned}$$
(2.9)
Then replace [1, (2.11)] by
$$\begin{aligned} \gamma ( \mathfrak {a}^*) = \sqrt{2} \mathfrak {a}^*\otimes 1_{\mathbb {C}^+}, \end{aligned}$$
(2.11)
and [1, (2.15)] by
$$\begin{aligned} \log \left( \Vert \mu \Vert ^{M,\nabla f}_{\det (H^\bullet (\widetilde{X}, \widetilde{F}), \mathbb {Z}_{2})}\right) (g)&= -\frac{1}{2} \log (2)\sum _{x\in B\cap Y_{1}} (-1)^{\mathrm{ind}(x)} \mathrm{rk}(F) \\&\quad +\! \log \Vert \gamma _{1}^{-1}\mu _{1}\Vert ^{M,\nabla f}_{\det H^{\bullet }(X, F)}\!+\!\chi (g) \log \Vert \gamma _{2}^{-1} \mu _{2}\Vert ^{M,\nabla f}_{\det H^{\bullet }(X, Y_{1},\, F)}. \end{aligned}$$
(2.15)
Finally, replace the bottom of p. 1100 from [1, (2.20)] on by:
“for \(\sigma _1\in H^\bullet (\widetilde{X}, \widetilde{F})^+\), \(\sigma _2\in H^\bullet (X, Y_1, F)\),
$$\begin{aligned}&P_\infty \circ \widetilde{\phi }_{1} \circ P_\infty ^{-1} (\sigma _1)|_{C_{\bullet }(W^u, F^{*})} = \frac{\sqrt{2}}{2} ( \sigma _1 + \phi ^{*} \sigma _1)|_{C_{\bullet }(W^u , F^{*})} = \gamma \sigma |_{C_{\bullet }(W^u, F^{*})},\nonumber \\&(P_\infty \circ \widetilde{\phi }_{2} \circ P_\infty ^{-1} \circ \gamma ) (\sigma _2\otimes 1_{\mathbb {C}^-} )|_{C_{\bullet }({W}^u/W^u_{Y_{1}}, \widetilde{F}^*)} = \sigma _2|_{C_{\bullet }({W}^u/W^u_{Y_{1}}, \widetilde{F}^*)}. \end{aligned}$$
(2.20)
Set
$$\begin{aligned} \tau _\pm = \gamma _\pm \circ P_\infty \circ \widetilde{\phi } \circ P_\infty ^{-1}\ : H^\bullet (\widetilde{X}, \widetilde{F})^{\pm }\rightarrow H^\bullet (\widetilde{X}, \widetilde{F})^{\pm }, \end{aligned}$$
(2.21)
with \(\gamma _+=\gamma _1, \gamma _-=\gamma _2\). By (2.20), we get
$$\begin{aligned} \tau _\pm =\mathrm{Id}.\text {''} \end{aligned}$$
(2.22)
