Examples

• Let ${\displaystyle M}$ be a compact, connected, oriented 2-dimensional manifold, i.e. a surface, of genus ${\displaystyle g}$ such that the boundary ${\displaystyle \partial M}$ is either empty or is connected. Embed ${\displaystyle M}$ in ${\displaystyle S^{m}}$, where ${\displaystyle m\geq 4}$. Choose a framing of M, that is a trivialization of the normal (m − 2)-plane vector bundle. (This is possible for ${\displaystyle m=3}$, so is certainly possible for ${\displaystyle m\geq 4}$). Choose a symplectic basis ${\displaystyle x_{1},x_{2},\ldots ,x_{2g-1},x_{2g}}$ for ${\displaystyle H_{1}(M)=\mathbb {Z} ^{2g}}$. Each basis element is represented by an embedded circle ${\displaystyle x_{i}:S^{1}\subset M}$. The normal (m − 1)-plane vector bundle of ${\displaystyle S^{1}\subset M\subset S^{m}}$ has two trivializations, one determined by a standard framing of a standard embedding ${\displaystyle S^{1}\subset S^{m}}$ and one determined by the framing of M, which differ by a map ${\displaystyle S^{1}\to SO(m-1)}$ i.e. an element of ${\displaystyle \pi _{1}(SO(m-1))\cong \mathbb {Z} _{2}}$ for ${\displaystyle m\geq 4}$. This can also be viewed as the framed cobordism class of ${\displaystyle S^{1}}$ with this framing in the 1-dimensional framed cobordism group ${\displaystyle \Omega _{1}^{\text{framed}}\cong \pi _{m}(S^{m-1})\,(m\geq 4)\cong \mathbb {Z} _{2}}$, which is generated by the circle ${\displaystyle S^{1}}$ with the Lie group framing. The isomorphism here is via the Pontrjagin-Thom construction. Define ${\displaystyle \mu (x)\in \mathbb {Z} _{2}}$ to be this element. The Arf invariant of the framed surface is now defined

${\displaystyle \Phi (M)=\operatorname {Arf} (H_{1}(M,\partial M;\mathbb {Z} _{2});\mu )\in \mathbb {Z} _{2}}$

Note that ${\displaystyle \pi _{1}(SO(2))\cong \mathbb {Z} ,}$ so we had to stabilise, taking ${\displaystyle m}$ to be at least 4, in order to get an element of ${\displaystyle \mathbb {Z} _{2}}$. The case ${\displaystyle m=3}$ is also admissible as long as we take the residue modulo 2 of the framing.

• The Arf invariant ${\displaystyle \Phi (M)}$ of a framed surface detects whether there is a 3-manifold whose boundary is the given surface which extends the given framing. This is because ${\displaystyle H^{1,1}}$ does not bound. ${\displaystyle H^{1,1}}$ represents a torus ${\displaystyle T^{2}}$ with a trivialisation on both generators of ${\displaystyle H_{1}(T^{2};\mathbb {Z} _{2})}$ which twists an odd number of times. The key fact is that up to homotopy there are two choices of trivialisation of a trivial 3-plane bundle over a circle, corresponding to the two elements of ${\displaystyle \pi _{1}(SO(3))}$. An odd number of twists, known as the Lie group framing, does not extend across a disc, whilst an even number of twists does. (Note that this corresponds to putting a spin structure on our surface.) Pontrjagin used the Arf invariant of framed surfaces to compute the 2-dimensional framed cobordism group ${\displaystyle \Omega _{2}^{\text{framed}}\cong \pi _{m}(S^{m-2})\,(m\geq 4)\cong \mathbb {Z} _{2}}$, which is generated by the torus ${\displaystyle T^{2}}$ with the Lie group framing. The isomorphism here is via the Pontrjagin-Thom construction.

• Let ${\displaystyle (M^{2},\partial M)\subset S^{3}}$ be a Seifert surface for a knot, ${\displaystyle \partial M=K:S^{1}\hookrightarrow S^{3}}$, which can be represented as a disc ${\displaystyle D^{2}}$ with bands attached. The bands will typically be twisted and knotted. Each band corresponds to a generator ${\displaystyle x\in H_{1}(M;\mathbb {Z} _{2})}$. ${\displaystyle x}$ can be represented by a circle which traverses one of the bands. Define ${\displaystyle \mu (x)}$ to be the number of full twists in the band modulo 2. Suppose we let ${\displaystyle S^{3}}$ bound ${\displaystyle D^{4}}$, and push the Seifert surface ${\displaystyle M}$ into ${\displaystyle D^{4}}$, so that its boundary still resides in ${\displaystyle S^{3}}$. Around any generator ${\displaystyle x\in H_{1}(M,\partial M)}$, we now have a trivial normal 3-plane vector bundle. Trivialise it using the trivial framing of the normal bundle to the embedding ${\displaystyle M\hookrightarrow D^{4}}$ for 2 of the sections required. For the third, choose a section which remains normal to ${\displaystyle x}$, whilst always remaining tangent to ${\displaystyle M}$. This trivialisation again determines an element of ${\displaystyle \pi _{1}(SO(3))}$, which we take to be ${\displaystyle \mu (x)}$. Note that this coincides with the previous definition of ${\displaystyle \mu }$.

• The Arf invariant of a knot is defined via its Seifert surface. It is independent of the choice of Seifert surface (The basic surgery change of S-equivalence, adding/removing a tube, adds/deletes a ${\displaystyle H^{0,0}}$ direct summand), and so is a knot invariant. It is additive under connected sum, and vanishes on slice knots, so is a knot concordance invariant.

• The intersection form on the (2k + 1)-dimensional ${\displaystyle \mathbb {Z} _{2}}$-coefficient homology ${\displaystyle H_{2k+1}(M;\mathbb {Z} _{2})}$ of a framed (4k + 2)-dimensional manifold M has a quadratic refinement ${\displaystyle \mu }$, which depends on the framing. For ${\displaystyle k\neq 0,1,3}$ and ${\displaystyle x\in H_{2k+1}(M;\mathbb {Z} _{2})}$ represented by an embedding ${\displaystyle x\colon S^{2k+1}\subset M}$ the value ${\displaystyle \mu (x)\in \mathbb {Z} _{2}}$ is 0 or 1, according as to the normal bundle of ${\displaystyle x}$ is trivial or not. The Kervaire invariant of the framed (4k + 2)-dimensional manifold M is the Arf invariant of the quadratic refinement ${\displaystyle \mu }$ on ${\displaystyle H_{2k+1}(M;\mathbb {Z} _{2})}$. The Kervaire invariant is a homomorphism ${\displaystyle \pi _{4k+2}^{S}\to \mathbb {Z} _{2}}$ on the (4k + 2)-dimensional stable homotopy group of spheres. The Kervaire invariant can also be defined for a (4k + 2)-dimensional manifold M which is framed except at a point.

• In surgery theory, for any ${\displaystyle 4k+2}$-dimensional normal map ${\displaystyle (f,b):M\to X}$ there is defined a nonsingular quadratic form ${\displaystyle (K_{2k+1}(M;\mathbb {Z} _{2}),\mu )}$ on the ${\displaystyle \mathbb {Z} _{2}}$-coefficient homology kernel

${\displaystyle K_{2k+1}(M;\mathbb {Z} _{2})=ker(f_{*}:H_{2k+1}(M;\mathbb {Z} _{2})\to H_{2k+1}(X;\mathbb {Z} _{2}))}$

refining the homological intersection form ${\displaystyle \lambda }$. The Arf invariant of this form is the Kervaire invariant of (f,b). In the special case ${\displaystyle X=S^{4k+2}}$ this is the Kervaire invariant of M. The Kervaire invariant features in the classification of exotic spheres by Michel Kervaire and John Milnor, and more generally in the classification of manifolds by surgery theory. William Browder defined ${\displaystyle \mu }$ using functional Steenrod squares, and C. T. C. Wall defined ${\displaystyle \mu }$ using framed immersions. The quadratic enhancement ${\displaystyle \mu (x)}$ crucially provides more information than ${\displaystyle \lambda (x,x)}$ : it is possible to kill x by surgery if and only if ${\displaystyle \mu (x)=0}$. The corresponding Kervaire invariant detects the surgery obstruction of ${\displaystyle (f,b)}$ in the L-group ${\displaystyle L_{4k+2}(\mathbb {Z} )=\mathbb {Z} _{2}}$.