The topology of a $k$-dimensional ball in $\mathbb R^N$ and the definition of attaching a $k$-cell

Question

Let $B_k$ be the $k$-dimensional ball in $\mathbb R^N$: $$ B_k = \left\{x \in \mathbb R^N \ : \ \sum_{i = 1}^k x_i^2 < 1, \ x_{k + 1} = \ldots = x_N = 0 \right\}. $$ In the context of Morse Theory, it is called a $k$-cell.

The operation of attaching a $k$-cell to a topological space $Y$ is defined in my book (Morse Index of Solutions of Nonlinear Elliptic Equations, Damascelli and Pacella, 2019) as follows:

Take a continuous function $g: \overline{\partial B_k} \to Y$ which is homeomorphic to its image. The topological space obtained by taking the union of $Y$ and the closed $k$-cell $\overline{B_k}$ with the equivalence relation which identifies $x \in \overline{\partial B_k}$ with its image $g(x) \in Y$ will be denoted by $Y \cup_g B_k$ ($Y$ with a $k$-cell attached).

Milnor, in his classical book, takes $g$ defined on $S^{k - 1}$

My questions are:

What is $\overline{\partial B_k}$, and why do we take it as the domain of $g$?

For the first part, as I understand, $\overline{\partial B_k} = \overline{B_k}$. Indeed, the boundary is obtained by taking the closure and excluding the interior. But the interior of $\overline{B_k}$ is empty. What am I missing?

What does it mean to take the equivalence relation $x \sim g(x)$ in $Y \cup_g B_k$? Intuitively, what are we doing?

I'm quite clueless about this.

Answer 1

If that is all what the authors say, it is poor notation.

1. One should emphasize that $B_k$ is topologically an open ball. But note that it is not an open subset of $\mathbb R^N$ unless $N = k$. Anyway, the closure $\overline{B_k}$ in $\mathbb R^N$ is always topologically a closed $k$-dimensional ball.

2. They authors do not define $\partial B_k$.

3. For $N > k$ the interpretation of $\partial B_k$ as the topological boundary of $B_k$ in $\mathbb R^N$ produces in fact $\partial B_k = \overline{B_k}$. Only for $N = k$ we get $\partial B_k = \overline{B_k} \setminus B_k$.

4. An interpretation of $\partial B_k$ which depends on $N$ does not make sense. Therefore the only reasonable interpretation is $\partial B_k = \overline{B_k} \setminus B_k$. But then $\partial B_k$ is closed and $ \overline{\partial B_k} = \partial B_k$. So it remains open why the authors write $\overline{\partial B_k}$. Anyway, $\partial B_k \approx S^{k-1}$. This agrees with Milnor.

5. The notation $Y \cup_g B_k$ is unusual because $g$ is defined on $\partial B_k$ which is disjoint from $B_k$. The standard notation is $Y \cup_g \overline{B_k}$