# Suppose $Y_f=D^n \cup_fY$ is the space obtained from $Y$ by attaching an n-cell via the map $f: S^{n-1} \rightarrow Y$, $n \geq 3$.

Suppose $$Y_f=D^n \cup_fY$$ is the space obtained from $$Y$$ by attaching an n-cell via the map $$f: S^{n-1} \rightarrow Y$$, $$n \geq 3$$.

I'm trying to understand the homology groups $$H_{n+1}(Y_f),H_{n}(Y_f),H_{n-1}(Y_f)$$ from the long exact sequence induced via mayer-vietoris sequence.

However, I am confused:

$$...\rightarrow H_{n+1}(Y_f) \rightarrow H_{n}(S^{n-1})\rightarrow H_{n}(Y) \oplus H_{n}(D^n) \rightarrow H_{n}(Y_f) \rightarrow H_{n-1}(S^{n-1}) \rightarrow H_{n-1}(Y) \oplus H_{n-1}(D^n) \rightarrow H_{n}(Y_f) \rightarrow...$$

Specifically on the map $$H_{n-1}(S^{n-1}) \rightarrow H_{n-1}(Y) \oplus H_{n-1}(D^n)$$. The Mayer Vietoris sequence is defined here by taking the generator, $$<1>$$, of $$H_{n-1}(S^{n-1}) \cong \mathbb{Z}$$ to $$(1,-1)$$.

My main question is this: What roll does the degree of the attaching map $$f$$ play in all of this? Shouldn't the degree of the attaching map show up in this homology sequence somehow? If not, why? If so, where? Thanks!

• What do you mean by 'degree' of $f$? The map represents a homotopy class in $\pi_{n-1}Y$. Are you assuming $Y$ is something in particular, for instance a sphere? – Tyrone Apr 20 at 9:16
• Maybe you think the homotopy type of the attaching map should come up because $S^{n-1}$ is in the exact sequence? It actually shouldn't be $S^{n-1}$, but rather the image of $S^{n-1}$ (and it shouldn't be $D^n$ but rather $D^n$ with its boundary identified under the attaching map). Images of functions give almost no information about the homotopy type of the function. For example, if the degree of a map $S^n \rightarrow S^n$ is not $0$, then the image is $S^n$. – Connor Malin Apr 20 at 10:20
• @Tyrone well the attaching map for the n-cell is a map from $S^{n-1}$ to $S^{n-1}$ and so has a degree associated with it ala how chain complexes are defined in cellular homology. – Mathematical Mushroom Apr 20 at 11:50
• Think about if I attach a disk through the constant map, the intersection will be just a point not a sphere. The attaching maps play an integral role in the homology, it is just the Mayer-Vietoris sequence doesn’t tell you it. Have you learned about cellular homology? It takes the exact approach you are looking for. – Connor Malin Apr 20 at 11:57
• @MathematicalMushroom, the space obtained from $Y$ by attaching an n-cell along a map $f:S^{n-1}\rightarrow Y$ is exactly that: the space obtained from $Y$ by attaching an n-cell along a map $f:S^{n-1}\rightarrow Y$. There is no map $S^{n-1}\rightarrow S^{n-1}$ here. The attaching map is the map $f:S^{n-1}\rightarrow Y$. – Tyrone Apr 20 at 12:25