Calculation of boundary of slant product in W. S. Massey's Singular Homology textbook In W.S.Massey's singular homology (Graduate Texts in Mathematics 70 Springer (1980)) there is a formula for the boundary of a double slant product on page 176
$$\partial(\phi\backslash\backslash a\otimes b\otimes g) = (\delta\phi)\backslash\backslash a\otimes b\otimes g + (-1)^{|\phi|}\phi\backslash\backslash\partial(a\otimes b\otimes g)$$
Am I right in calculating that this is missing a minus sign and should be
$$\partial(\phi\backslash\backslash a\otimes b\otimes g) = -(\delta\phi)\backslash\backslash a\otimes b\otimes g + (-1)^{|\phi|}\phi\backslash\backslash\partial(a\otimes b\otimes g)?$$
If the minus sign is required here, then it is surely also required subsequently, for example in the formula
$$\partial(u\backslash v)=(\delta u)v+(-1)^{|u|}u\backslash\partial(v)$$
which should then be
$$\partial(u\backslash v)=-(\delta u)v+(-1)^{|u|}u\backslash\partial(v).$$
I have been for some years puzzled by this as the formulae in the book give the appearance of having signs in the expected places. I would be very happy to be put right on this issue.
 A: You and Tyrone are correct that this is a sign mismatch.
Just to be clear: Massey writes this sign for the double slant product because it is the desirable one. It would imply that we have a map of chain complexes
$$C^*(Y, G_1) \otimes C_*(X) \otimes C_*(Y) \otimes G_2 \to C_*(X) \otimes G_1 \otimes G_2$$
where this is the tensor product of chain complexes, using the Koszul sign convention.
This is not compatible with the other things that Massey writes. In particular, a couple of lines later he writes that we should take the convention
$$(\delta \phi)(b) = (-1)^{|\phi|} \phi(\partial b)$$
(which does not alter the cohomology groups) and this is the source of the difficulty. This definition does not make the pairing
$$C^*(Y, G_1) \otimes C_*(Y) \to G_1$$
into a chain map.
Many people (myself included) would prefer to use the sign convention
$$(\delta \phi)(b) = (-1)^{|\phi|+1} \phi(\partial b)$$
for the coboundary operator $\delta$ precisely to fix this type of problem. This looks like it violates the Koszul sign convention, but duality has a tendency to do that. If you make this change, I believe Massey's sign for the double slant product works out.
(One possible explanation is this. In the definition of $\partial$ you take an alternating sum of faces, starting at the 0'th and ending at the n'th. The sign convention I've written above for $\delta$ on cochains corresponds to taking an alternating sum over faces, starting at the n'th and ending at the 0'th. Duality tends to interact better with this order reversal of the faces.)
