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!

  • 1
    $\begingroup$ 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? $\endgroup$ – Tyrone Apr 20 at 9:16
  • 1
    $\begingroup$ 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 $S^n$. $\endgroup$ – Connor Malin Apr 20 at 10:20
  • $\begingroup$ @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. $\endgroup$ – Mathematical Mushroom Apr 20 at 11:50
  • 1
    $\begingroup$ 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. $\endgroup$ – Connor Malin Apr 20 at 11:57
  • 1
    $\begingroup$ @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 $f:S^{n-1}\rightarrow Y$. $\endgroup$ – Tyrone Apr 20 at 12:25

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.