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In a book I read:

$M$ and $N$ are $d$-dimensional manifolds. Let $\widetilde{M}$ be the $(d−1)$-skeleton of $M$, or equivalently, $\widetilde{M}$ is obtained from $M$ by removing a disc in the interior of the $d$-cell of $M$. Define $\widetilde{N}$ similarly. Suppose that $f: S^{d−1} \to \widetilde{M}$ and $g: S^{d−1} \to \widetilde{N}$ are the attaching maps for the top cells in $M$ and $N$. Then the attaching map for the top cell in the connected sum $M\#N$ is $S^{d−1}\xrightarrow{f+g} \widetilde{M}\vee\widetilde{N}$.

So far, the denotation $f+g$ looks like just a denotation, however later the book deals with a specific example of manifold $S^3\times S^4$, where the attaching map $S^6 \to S^3 \vee S^4$ for its top cell is the Whitehead product $[s_1, s_2]_w$, where $s_1$ and $s_2$ respectively are the inclusions of $S^3$ and $S^4$ into $S^3 \vee S^4$. The attaching map for the top cell of the connected sum $(S^3 \times S^4)\#(S^3 \times S^4)$ is therefore the sum of two such Whitehead products. Finally, it is said that passing to the adjoint map we obtain $[a_1,b_2]+[a_2,b_2]$ for $a_i,b_i\in H_*(\Omega X).$

There are specific $a_i,b_i$ offered in the book, but I don't think it is very relevant to the question.

So, my question is: where does the denotation $f+g$ come from? Does this map has some kind of a property of transforming into a real sum when passing to adjoint maps?

Haldot
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    The notation $f+g$ comes from the fact that there is an external addition of maps $f+g: X \vee Y \rightarrow W \vee Z$ for map $f: X \rightarrow W$, $g: Y \rightarrow Z$ given by doing the respective map on each wedge summand. This is "addition" as much as the operation in homotopy groups is addition since the operation on homotopy groups comes from setting $X = S^n$ and precomposing with the map $S^n \rightarrow S^n \vee S^n$. – Connor Malin May 18 '20 at 02:09
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    I will also add that the paragraph you quote is very sloppy (unless I am somehow missing something). One has to be very careful in removing subsets from a CW complex and saying we still have a CW complex. First off, the subset should be a cell (which it is not in the quote). If it is not a subset, one should argue that the CW structure can be manipulated to make it one (perhaps this can be done here; I'm not sure$. Second, there is no mention of basepoints. They should be taken to be on the boundary of the disk removed. – Connor Malin May 18 '20 at 02:13
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    Third, we should not take the attaching map of the top cell (this makes no sense). We will not get a manifold. Instead we should glue a cylinder along the inclusion of the boundary into the disjoint union of the manifolds with the removed disks. – Connor Malin May 18 '20 at 02:15
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    See here for an example if the 2-dimensional case. The idea of what's going on in general should be clear. – Tyrone May 18 '20 at 08:45

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