Understanding the statement of a question.

Question

The question is given below:

Let $f : S^n \rightarrow S^n$ be a map of degree $m.$ Let $ X=S^n \cup_{f} D^{n+1}$ be the space attained from $S^n$ by attaching an $(n+1)$-cell via $f.$ Compute homology groups of $X.$

My questions are:

1- What is the importance of our map to be of degree m? what is the relation between $m$ and $n$?

2-Why when attaching an $(n+1)$-cell via $f$ that means we are attaching D ^{n+1} ?

3- Is the union supposed to be a disjoint union?

Answer 1

Regarding question 1, $m$ and $n$ are independent parameters. What you are asked to do is to answer this question for all $(m,n) \in \mathbb N \times \mathbb N$.

Regarding questions 2 and 3, what the expression $$X = S^n \cup_f D^{n+1} $$ means is that $X$ is the quotient of the disjoint union $S^n \coprod D^{n+1}$, using the decomposition which identifies each $x \in S^n = \partial D^{n+1} \subseteq D^{n+1}$ (the second term of the disjoint union) with the point $f(x) \in S^n$ (the first term of the disjoint union).

Answer 2

This should answer 2 and 3.

$X=S^n\cup_f D^{n+1}$ means, by definition that $$X=\big(\text{disjoint unionion of }S^n\text{ and }D^{n+1}\big)/R$$ where $R$ is an equivalence relation given by: $xRy\iff f(x)=y$ for $x\in S^n$ and $y\in \partial D^{n+1}(=S^n)$.