# Different definitions of the linking number

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.

• I don't have the reference here, but I believe it's all spelled out in Bredon's Geometry and Topology textbook. I imagine someone will stop by with a precise reference soon. In the text he build the links between degrees intersection theory, Thom classes, Poincare duality, etc. – Ryan Budney Feb 27 at 22:10
• Is Proposition 3.3 in DeTurck and Gluck's "Linking Integrals in the n-sphere" explicit enough? mat.unb.br/~matcont/34_10.pdf The main focus of the paper is on an integral formula for the linking number though. – j.c. Mar 18 at 16:44
• @j.c. Thank you for the reference. I was not aware of this paper. If you post it as an answer, I will accept it. – Piotr Hajlasz Mar 18 at 16:57

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).