The Pontryagin–Thom construction in algebraic geometry
Count the points in a fiber. Attach a sign to each. In algebraic geometry the sign becomes a unit, the count becomes a quadratic form, and the relations come from the zeros and poles of rational functions.
How many ways can a circle wrap a circle?
A map $f\colon S^1\to S^1$ has a winding number: how many net times the image goes around.
To compute it: pick a regular value $c$ on the target, look at its preimage, and count the points with signs, + where $f$ passes through $c$ forwards and − where it passes backwards.
The signed count does not depend on $c$, and it does not change if you wiggle $f$. It is a homotopy invariant, and it is a complete one: $[S^1,S^1]\cong\mathbb Z$.
Preimages come and go in ± pairs
The graph is a map $S^1\to S^1$ drawn on a square with opposite sides glued. Dashed line: the target value $c$. Marked points: its preimage.
Every time a new pair of preimages is born, one is + and one is −. That is the whole reason the signed count is invariant.
The signed count is the whole story, in every dimension
For a smooth map $f\colon S^{n+k}\to S^k$ and a regular value $c$, the preimage $f^{-1}(c)$ is an $n$-manifold, and it comes with a framing: a trivialization of its normal bundle, pulled back from the target.
Conversely, a framed submanifold gives a map: collapse everything outside a tube around it to the point at infinity.
Homotopy classes of maps $S^{n+k}\to S^k$, for $k$ large, correspond to framed $n$-manifolds up to framed cobordism. For $n=0$ these classes form $\pi_0(\mathbb S)$, the zeroth stable homotopy group, and they are finite sets of signed points modulo canceling ± pairs: $\pi_0(\mathbb S)\cong\mathbb Z$.
With a group acting, the answer is the Burnside ring
Let a finite group $G$ act on everything. A fiber is now a finite $G$-set: a disjoint union of orbits $G/H$.
The Burnside ring $A(G)$: finite $G$-sets, with $\sqcup$ as addition and $\times$ as multiplication, and formal differences allowed.
$\pi_0^G(\mathbb S)\cong A(G)$, and the isomorphism is Pontryagin–Thom: send a $G$-set to the collapse map of its orbits.
How do you tell two $G$-sets apart? Count fixed points. The marks $S\mapsto |S^H|$, one for each subgroup $H$ up to conjugacy, determine $S$ (Burnside).
Five stops
The plan for the rest of the talk: do exactly this, with polynomials instead of manifolds. Stops S4 and S5 are where it gets interesting.
Now do it with polynomials
Three things change
1. Homotopies are polynomial. A deformation is a family $f_t$ with $t$ a polynomial variable. There is no such thing as a small perturbation: $t$ ranges over the whole line.
2. Points have fields. Over $\mathbb Q$, the equation $x^2=2$ cuts out a single point, of degree 2, with residue field $\mathbb Q(\sqrt2)$. It becomes two points once $\sqrt2$ is available, e.g. over $\mathbb R$.
3. Points can be fat. $x^2=0$ is one point of length 2. A fiber of a polynomial map can be non-reduced, and no perturbation is automatically available to fix it.
A framed point, algebraically
A framed correspondence (Voevodsky) is a triple $(Z,U,\varphi)$: a finite set of (possibly fat) points $Z\subset\mathbb A^n$, a neighborhood $U$, and $n$ equations $\varphi$ on $U$ whose zero set is exactly $Z$.
Here $\varphi=(x+2)(x^2-2)$ on $U=\mathbb A^1\smallsetminus\{-2\}$, so $Z=V(x^2-2)$ $=\operatorname{Spec}\mathbb Q(\sqrt2)$.
The extra zero at $-2$ is thrown away by shrinking $U$. That freedom is what makes these different from plain polynomial maps.
Why these? Voevodsky's idea, now a theorem of Garkusha–Panin (see also Elmanto–Hoyois–Khan–Sosnilo–Yakerson): for $k$ infinite and perfect, $\pi_0(\mathbb S_k)(k)\cong\pi_0^{\mathrm{fr}}(k)$, the group of framed correspondences up to $\mathbb A^1$-homotopy. This is the algebraic Pontryagin–Thom theorem.
The sign becomes a unit
In topology the framing at a point is a sign: the slope of $f$ at the preimage is positive or negative.
Algebraically the slope is an actual number in the residue field: $\varphi'(\sqrt2)=4(1+\sqrt2)$. The theorem will show that only its class up to squares matters; $4$ is a square, so this is the unit $1+\sqrt2$.
So a framed point is a pair: a field $L$ and a unit $u\in L^\times$. Over $\mathbb R$ that is the sign again.
The unit-decorated Burnside ring $A^u(\Gamma_k)$, where $\Gamma_k=\mathrm{Gal}(\bar k/k)$ plays the role of $G$: finite étale $k$-schemes with a unit, under $\sqcup$ and $\times$, with formal differences. Generators $(L,u)$: $L/k$ finite separable, $u\in L^\times$.
Counting with units: quadratic forms
A quadratic form over $k$: a homogeneous quadratic polynomial, like $2x^2+8xy+4y^2$, up to invertible linear substitution of the variables.
Write $\langle a_1,\dots,a_n\rangle$ for $a_1x_1^2+\dots+a_nx_n^2$. Substituting $x\mapsto cx$ ($c\ne0$) shows $\langle u\rangle=\langle uc^2\rangle$, and completing the square diagonalizes every form (for $\operatorname{char}k\ne2$).
Adding forms in separate variables, multiplying by $\langle a\rangle\langle b\rangle=\langle ab\rangle$, and allowing formal differences gives the Grothendieck–Witt ring $\mathrm{GW}(k)$. The hyperbolic plane is $h=\langle1\rangle+\langle-1\rangle$, the form $x^2-y^2$.
| field | GW(k) | what it counts |
|---|---|---|
| ℂ | ℤ | points |
| ℝ | ℤ ⊕ ℤ | rank and signature |
| 𝔽q, q odd | ℤ ⊕ ℤ/2 | rank and discriminant |
| ℚ | big | rank, signature, discriminant, Hasse invariants at every prime |
Over ℝ, the signature is exactly the signed count from topology. It is also what separates the ellipse from the hyperbola in the curves $q=1$ (left): a substitution can stretch one but never turn it into the other. The rest of the table is what the algebra remembers and topology forgets.
Counting a framed point
A framed point $(L,u)$ is counted by the Dress map $$d(L,u)=\operatorname{tr}_{L/k}\langle u\rangle:\qquad z\ \longmapsto\ \operatorname{tr}_{L/k}\big(uz^2\big),$$ a quadratic form on $L$, viewed as a $k$-vector space.
Over $\mathbb R$: signature $0$, matching the slopes $+9.66$ and $-1.66$. Over $\mathbb Q$: not hyperbolic, since $2$ is not a square. The algebra remembers more than the sign.
One irrational point splits into two rational ones
$\varphi_t=(x+2)\big(x^2+tx+2t-2\big)$. At $x=-2$ the quadratic factor equals $2$ for every $t$, so no zero reaches the excised point; the leading coefficient is $1$, so no zero escapes to infinity.
The picture shows real points and their slopes $u$ (at $t=0$ these are $4(1\pm\sqrt2)$; $4$ is a square). The form is written in the basis $1,\theta$, with $\theta$ a root. Over $\mathbb Q$ the fiber is one point unless $t^2-8t+8$ is a square.
Segal's theorem over a field
For $k$ an infinite perfect field of characteristic $\ne2$, the Pontryagin–Thom map $$\mathrm{PT}\colon A^u(\Gamma_k)\longrightarrow\pi_0(\mathbb S_k)(k)$$ which sends $(L,u)$ to the class of a framed point with residue field $L$ and framing determinant $u$, is a surjective ring homomorphism with kernel $\ker(d)$. Hence $$\pi_0(\mathbb S_k)(k)\;\cong\;A^u(\Gamma_k)/\ker(d)\;\cong\;\mathrm{GW}(k).$$
$\pi_0(\mathbb S_k)(k)\cong\mathrm{GW}(k)$ is Morel's theorem (2004; reproved in his 2012 book), via strictly $\mathbb A^1$-invariant sheaves and Milnor–Witt $K$-theory. Neshitov and Druzhinin–Kylling gave framed-correspondence proofs. New here: Segal's architecture, the Burnside source $A^u(\Gamma_k)$, and the kernel.
Same shape, but the isomorphism has moved one step to the right (the composite is the marks, resp. the Dress map $d$). PT now has a kernel, generated by principal quadratic divisors: zeros and poles of rational functions on the line.
Under the hood
Every ingredient has a translation
S1–S3: $e(L,u)$, the class of any framed point with support $\operatorname{Spec}L$ and framing determinant $u$, is well defined, and $\mathrm{PT}(L,u)=e(L,u)$ is a ring map (details on a backup slide).
Framed recognition (slide 9) is a black box. Precisely, $\pi_0^{\mathrm{fr}}(k)=\mathbb ZF(k)/\mathbb A^1$-homotopy, where $\mathbb ZF(k)=\operatorname{colim}_n\mathbb ZF_n(k)$: formal sums of level-$n$ triples $(Z\subset\mathbb A^n,U,\varphi)$ with $Z'\sqcup Z''=Z'+Z''$, stabilized by $(Z,U,\varphi)\mapsto(Z\times0,\,U\times\mathbb A^1,\,(\varphi,t))$. An $\mathbb A^1$-homotopy is a framed correspondence over $\mathbb A^1_k$ whose support is finite over $\mathbb A^1$.
A quadratic form from a framed point
For $(Z,U,\varphi)$ with $Z$ finite, the algebra $\mathcal O(Z)=\mathcal O(U)/(\varphi)$ is a complete intersection. Scheja–Storch build a canonical functional $\eta\colon\mathcal O(Z)\to k$ with $\eta(\det d\varphi)=\dim_k\mathcal O(Z)$, and the form $$w(Z,U,\varphi)\;=\;\big[(a,b)\mapsto\eta(ab)\big]\ \in\ \mathrm{GW}(k)$$ is nondegenerate.
Harder's theorem $\mathrm{GW}(k[t])=\mathrm{GW}(k)$ makes $w$ invariant under $\mathbb A^1$-homotopy, so $w\colon\pi_0^{\mathrm{fr}}(k)\to\mathrm{GW}(k)$.
$w\circ\mathrm{PT}=d$: on an étale point, $w(e(L,u))=\operatorname{tr}_{L/k}\langle u\rangle$.
This is the analogue of Segal's fixed-point degree. There it is elementary; here serious duality theory enters, and it is what lets us avoid every prior computation of $\pi_0$.
For $k=\mathbb R$ the signature of $w$ is the sum of the local degrees at the real points of $Z$ (Eisenbud–Levine, Khimshiashvili).
Surjectivity of $w$ is immediate: $w(e(k,v))=\langle v\rangle$. Injectivity is the content of S4 and S5.
Fatness carries no information
Every class in $\pi_0^{\mathrm{fr}}(k)$ is a $\mathbb Z$-combination of classes with finite étale support. So PT is onto. (Neshitov, characteristic 0; Druzhinin–Kylling, perfect fields; implicit over any field in BEHKSY.)
Why it is not free: a naive perturbation has zeros that escape to infinity, and then the family is not an $\mathbb A^1$-homotopy. Example: $\varphi=(x^2+x^2y,\ y^2)$ cuts out a fat point of length 4; its fiber over $(1,1)$ has length 2, the other two points having gone to infinity.
Two proofs. Moving via an étale correspondence (Levine; BEHKSY): Bertini in the framed Hilbert scheme, which is smooth where the Hilbert scheme of points is not. Re-topping: inside a complete-intersection ideal, Nakayama lets you replace the framing by one with leading forms $C\cdot(x_1^M,\dots,x_n^M)$, $C$ invertible; such a map is finite flat, so its deformation is proper over the whole line by construction.
$y=sx^2+x-1$: as $s\to0^+$ one zero runs off to $-\infty$.
The framing is essential: an unframed fat point can degenerate to a square-zero scheme, which would collapse $\pi_0$ to the rank (this is how $\mathrm{Hilb}(\mathbb A^\infty)$ computes $K$-theory). That degeneration leaves the complete intersections, so framed points cannot use it.
Every étale point splits into rational ones
Cazanave (2012): naive homotopy classes of pointed rational functions $A/B$ ($A$ monic of degree $n$, $\deg B\lt n$) are classified by the Bézout form, the symmetric $n\times n$ matrix of $\dfrac{A(x)B(y)-A(y)B(x)}{x-y}$, together with the resultant $\operatorname{Res}(A,B)\in k^\times$.
The framed point $e(L,u)$ is the class of $A/B$, with $A$ the minimal polynomial and $B(\theta)=A'(\theta)/u$. Its Bézout form is $\operatorname{tr}_{L/k}\langle u\rangle$. Matching Bézout form and resultant (rescale one unit by a square to fix the resultant) deforms it to a split function: distinct rational zeros, with units $v_1,\dots,v_n$.
$e(L,u)=\sum_i e(k,v_i)$ whenever $\operatorname{tr}_{L/k}\langle u\rangle\cong\langle v_1,\dots,v_n\rangle$. So $s\langle v\rangle:=e(k,v)$ is a well-defined section of $w$, and $\mathrm{PT}=s\circ d$.
Endgame, exactly as in Segal:
- $w\circ s=\mathrm{id}$ (Scheja–Storch on rational points).
- $s\circ w=\mathrm{id}$ on étale classes, and by S4 every class is étale.
- So $w$ and $s$ are inverse, $\pi_0^{\mathrm{fr}}(k)\cong\mathrm{GW}(k)$, and $\ker\mathrm{PT}=\ker(w\circ\mathrm{PT})=\ker d$.
Demo 2 was an instance (its unit was $4(1+\sqrt2)$, and $4$ is a square): $e(\mathbb Q(\sqrt2),1+\sqrt2)=e(\mathbb Q,2)+e(\mathbb Q,-1)$.
What does the Dress map forget?
Segal's marks are injective. The Dress map $d\colon A^u(\Gamma_k)\to\mathrm{GW}(k)$ is surjective and far from injective.
Classical. A rational function on $\mathbb P^1$ has as many zeros as poles, counted with multiplicity and degree.
$\ker(d)$ is generated by the principal quadratic divisors $R(f)$, one for each rational function $f$ on $\mathbb P^1_k$: its finite odd-order zeros and poles with their slopes, minus $\infty$ with its slope if of odd order, minus $r$ copies of $h_A=(k,1)+(k,-1)$.
So: $\pi_0(\mathbb S_k)(k)\cong A^u(\Gamma_k)\,/\,$principal quadratic divisors.
The Burnside source has to be cut down
Segal's map $A(G)\to\pi_0^G(\mathbb S)$ is an isomorphism. Ours is onto, with a kernel. Why?
Segal uses only canonical framings. A point with another framing is not a new generator; it is expressed through the others. Example: $C_2$ acting on $\mathbb R$ by $x\mapsto-x$. The fixed point with the opposite framing is $[\mathrm{pt}]-[C_2/e]$; both have marks $(-1,1)$.
$A^u(\Gamma_k)$ keeps every unit as a separate generator, so it must be cut down. The kernel records how a point with an arbitrary unit is expressed through others. Those expressions come from families that change the type of the points, as in Demo 2, and they are exactly the principal quadratic divisors.
The reading of the theorem. Quadratic forms over $k$ are what you get from finite étale $k$-schemes with units, if you let points split and merge along rational functions on the line.
Demo 1 in these terms: a ± pair of real preimages that is born or dies is a complex-conjugate pair landing on the real line and splitting in two, or the reverse.
Over $\mathbb Q$: rank, signature, discriminant and Hasse invariants at every prime, and every relation comes from a rational function on $\mathbb P^1_{\mathbb Q}$.
What this suggests
- Higher weights. Morel: $\pi_{n,n}(\mathbb S_k)\cong K^{MW}_{-n}(k)$. Is there a Morel-independent Burnside-style presentation, by decorated finite étale data and divisor relations?
- Reconstruction. Genuine $G$-spectra are spectral Mackey functors on the Burnside category (Guillou–May). Conjecture: the heart of the Artin part of $\mathcal{SH}(k)$ is additive presheaves on a Milnor–Witt span category of finite étale $k$-schemes.
- Beyond a field. Over a base scheme, or with a group acting, the Burnside source is still canonical; the moving lemmas are the problem.
Count the points. Keep the units. Divide by principal quadratic divisors.
Partially supported by NSF award DMS-2604878.Bibliography
Topology
L. S. Pontryagin, Smooth manifolds and their applications in homotopy theory, Trudy Mat. Inst. Steklov. 45 (1955); Amer. Math. Soc. Transl. (2) 11 (1959), 1–114.
R. Thom, Quelques propriétés globales des variétés différentiables, Comment. Math. Helv. 28 (1954), 17–86.
G. Segal, Equivariant stable homotopy theory, Actes du Congrès International des Mathématiciens (Nice, 1970), Tome 2, Gauthier-Villars, 1971, 59–63.
T. tom Dieck, Transformation groups and representation theory, Lecture Notes in Math. 766, Springer, 1979.
The motivic sphere
F. Morel, On the motivic π0 of the sphere spectrum, in Axiomatic, enriched and motivic homotopy theory, NATO Sci. Ser. II 131, Kluwer, 2004, 219–260.
F. Morel, 𝔸1-algebraic topology over a field, Lecture Notes in Math. 2052, Springer, 2012.
Quadratic forms and degrees
A. W. M. Dress, Notes on the theory of representations of finite groups, Universität Bielefeld, 1971.
W. Scharlau, Quadratic reciprocity laws, J. Number Theory 4 (1972), 78–97.
T. Y. Lam, Introduction to quadratic forms over fields, Grad. Stud. Math. 67, AMS, 2005 (Ch. VII for Harder's theorem).
G. Scheja and U. Storch, Über Spurfunktionen bei vollständigen Durchschnitten, J. Reine Angew. Math. 278/279 (1975), 174–190.
D. Eisenbud and H. I. Levine, An algebraic formula for the degree of a C∞ map germ, Ann. of Math. (2) 106 (1977), 19–44.
G. N. Khimshiashvili, On the local degree of a smooth mapping, Soobshch. Akad. Nauk Gruzin. SSR 85 (1977), 309–312.
C. Cazanave, Algebraic homotopy classes of rational functions, Ann. Sci. Éc. Norm. Supér. 45 (2012), 511–534.
Framed correspondences
V. Voevodsky, Notes on framed correspondences, unpublished, 2001.
G. Garkusha and I. Panin, Framed motives of algebraic varieties (after V. Voevodsky), J. Amer. Math. Soc. 34 (2021), 261–313.
E. Elmanto, M. Hoyois, A. A. Khan, V. Sosnilo, and M. Yakerson, Motivic infinite loop spaces, Camb. J. Math. 9 (2021), 431–549.
T. Bachmann, E. Elmanto, M. Hoyois, A. A. Khan, V. Sosnilo, and M. Yakerson, On the infinite loop spaces of algebraic cobordism and the motivic sphere, Épijournal Géom. Algébrique 5 (2021), Art. 1.
A. Neshitov, Framed correspondences and the Milnor–Witt K-theory, J. Inst. Math. Jussieu 17 (2018), 823–852.
A. Druzhinin and J. I. Kylling, Framed correspondences and the zeroth stable motivic homotopy group in odd characteristic, arXiv:1809.03238.
M. Hoyois, J. Jelisiejew, D. Nardin, B. Totaro, and M. Yakerson, The Hilbert scheme of infinite affine space and algebraic K-theory, arXiv:2002.11439.
AI use. AI assistants — Claude Fable 5.1 and Opus 5.5 — were used to help draft these slides and develop parts of the mathematics in this project (especially the moving lemmas for framed correspondences). The author takes full responsibility for the mathematical content.
The map, and why it is well defined
S1 (collapse). For $L=k(\theta)$ with minimal polynomial $A$ and $u\in L^\times$, let $e(L,u)$ be the class of $\big(V(A),\ \mathbb A^1\smallsetminus V(B),\ A/B\big)$ with $B(\theta)=A'(\theta)/u$, whose framing determinant at $\theta$ is $A'(\theta)/B(\theta)=u$. Set $\mathrm{PT}(L,u)=e(L,u)$.
S2 (isotopy). Framing rigidity: two framings of the same étale $Z$ with the same Jacobian determinant on $Z$ give the same class. Shears, $SL_n$, and simultaneous changes of coordinates act trivially, so the class depends only on $\det d\varphi|_Z\in\mathcal O(Z)^\times$.
S3 (ring). $\sqcup\mapsto+$, and the external product of $(L,u)$ and $(L',u')$ has support $\operatorname{Spec}(L\otimes_kL')$ with unit $u\otimes u'$. PT is a ring map.
What survives. The homotopy $(\varphi_1(1+t\nu),\varphi_2,\dots)$, $\nu$ vanishing on $Z$, is free, so only the value of $\det d\varphi$ on $Z$ matters: a unit in $\mathcal O(Z)^\times$. Nothing here identifies $u$ with $uc^2$; that happens at S5, and it takes a second point.
The substitute for “$GL_n^+(\mathbb R)$ is connected”: $SL_n=E_n$ over semilocal rings, and elementary matrices are $\mathbb A^1$-homotopic to the identity.
Residues, the Euler unit, and $R(f)$
Points. A closed point $p\in\mathbb A^1_k$ has monic irreducible $\pi_p\in k[x]$, residue field $\kappa(p)=k[x]/(\pi_p)$, and root $\theta_p$. At $\infty$ take $\pi_\infty=1/x$ and $\kappa(\infty)=k$.
Euler unit. $\varepsilon_p:=\pi_p'(\theta_p)\in\kappa(p)^\times$, the slope of the minimal polynomial at its root.
Leading unit. Write $f=\pi_p^{v_p(f)}g_p$ with $g_p$ a unit at $p$, and set $\ell_p(f):=g_p(p)\in\kappa(p)^\times$.
Second residue. $\partial_p\colon\mathrm W(k(x))\to\mathrm W(\kappa(p))$, $\ \partial_p\langle\pi_p^n g\rangle=\langle g(p)\rangle$ for $n$ odd, $0$ for $n$ even.
Scharlau's reciprocity law (1972). For every $q\in\mathrm W(k(x))$, $$\sum_{p\in(\mathbb A^1_k)^{(1)}}\operatorname{tr}_{\kappa(p)/k}\big(\langle\varepsilon_p\rangle\otimes\partial_pq\big)\;=\;\partial_\infty q\quad\text{in }\mathrm W(k).$$ The Euler twist is essential: for $q=\langle x^2-2\rangle$ it turns the trace form $\langle1,2\rangle\neq0$ into $\operatorname{tr}\langle\sqrt2\rangle$, which is hyperbolic.
Quadratic divisor. For $f\in k(x)^\times$, $$\begin{aligned}\rho(f):=&\sum_{\substack{p\neq\infty\\ v_p(f)\text{ odd}}}\big(\kappa(p),\,\varepsilon_p\,\ell_p(f)\big)\\&-\;[v_\infty(f)\text{ odd}]\,\big(k,\,\ell_\infty(f)\big)\ \in A^u(\Gamma_k).\end{aligned}$$ At a simple zero $\varepsilon_p\ell_p(f)=f'(\theta_p)$; at a simple pole it is $(1/f)'(\theta_p)$ up to squares. These are the slopes of the kernel slide.
Rank correction. $\deg\operatorname{div}(f)=0$ makes $\operatorname{rank}\rho(f)=2r(f)$ even. The principal quadratic divisor of $f$ is $R(f):=\rho(f)-r(f)\,h_A$, with $h_A=(k,1)+(k,-1)$.
For $k$ perfect of characteristic $\neq2$ (infinite not needed here), $\ker(d)$ is the subgroup generated by the $R(f)$.
Proof outline. $R(f)\in\ker d$: apply Scharlau's law to $q=\langle f\rangle$; then $d(\rho(f))=0$ in $\mathrm W(k)$, so it is $r(f)h$ in $\mathrm{GW}(k)$. $\ker d\subseteq\langle R(f)\rangle$: every class is congruent mod $R$ to one on rational points, and Witt's relations among the $\langle u\rangle$ are themselves of the form $R(f)$.