Different definitions of the linking number

Question

Assume that $$ \iota_1:\mathbb{S}^k\to\mathbb{R}^n, \quad \iota_2:\mathbb{S}^\ell\to\mathbb{R}^n, \quad k+\ell=n-1, $$ are two embeddings of spheres with disjoint images $\iota_1(\mathbb{S}^k)\cap\iota_2(\mathbb{S}^\ell)=\emptyset$. Then, the linking number $Lk(\iota_1,\iota_2)$ of the embedded spheres can be defined as the degree of the mapping $$ \mathbb{S}^k\times\mathbb{S}^\ell\ni (x,y)\longmapsto \frac{\iota_1(x)-\iota_2(y)}{\Vert \iota_1(x)-\iota_2(y)\Vert} \in \mathbb{S}^{n-1}. $$ There is however, another approach.

By the Alexander duality $H_k(\mathbb{R}^n\setminus \iota_2(\mathbb{S}^\ell))=\mathbb{Z}$ and the embedding $\iota_1$ induces a homomorphism of the homology groups: $$ \iota_{1*}:\underbrace{H_k(\mathbb{S}^k)}_{\mathbb{Z}}\to H_k(\mathbb{R}^n\setminus\iota_2(\mathbb{S}^\ell))=\mathbb{Z}. $$ Thus, the image of a generator of $H_k(\mathbb{S}^k)$ is an integer.

It is well known that this integer equals (up to a sign) to the linking number $Lk(\iota_1,\iota_2)$.

While this is a well known result, I could not find any reference for a proof in a book.

Question. Do you know any reference where this result is stated and proved explicitly?

I find is quite surprising that except for the case of links and knots (dimension one) there are almost no references for the linking number.

The point is that the linking number is often used by researchers and analysis and geometry who have a limited knowledge in algebraic topology (speaking of myself) and a straightforward reference would be of a great help.

Answer 1

Proposition 3.3 in De Turck and Gluck's "Linking Integrals in the n-sphere" states:

Let $K^k$ and $L^\ell$ be disjoint closed oriented smooth submanifolds of $S^n$ with $k+\ell=n-1$ and let $f:K^k\ast L^\ell \rightarrow S^n$ be [the map sending the line segment $\{(\mathbf{x}, \mathbf{y}, u)\mid0\leq u \leq 1\}$ connecting $\mathbf{x}$ and $\mathbf{y}$ in $K\ast L$ proportionally to the geodesic arc connecting $\mathbf{x}$ and $-\mathbf{y}$ in $S^n$]. Then

$$\deg f = - \operatorname{Lk}(K^k, L^\ell).$$

In this paper, the linking number $\operatorname{Lk}$ is defined as an intersection number (Eq. 2.1) (thus, related to the definition you gave using homology).