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Kirsten Wickelgren

Publications and source records attributed to Kirsten Wickelgren.

At least 37 records · Page 2Linked to original sources
arXiv2020

An Example of Wild Ramification in an Enriched Riemann-Hurwitz Formula

M. Levine proved an enrichment of the classical Riemann-Hurwitz formula to an equality in the Grothendieck-Witt group of quadratic forms. In its strongest form, Levine's theorem includes a technical hypothesis on ramification relevant in positive characteristic. We consider wild ramification at points whose residue fields are non-separable extensions of the ground field k. We show an analogous Riemann-Hurwitz formula, and consider an example suggested by S. Saito.

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arXiv2018

The Simplicial EHP Sequence in A1-Algebraic Topology

We give a tool for understanding simplicial desuspension in A1-algebraic topology: we show that $X \to Ω(S^1 \wedge X)\to Ω(S^1 \wedge X \wedge X)$ is a fiber sequence up to homotopy in 2-localized A1-algebraic topology for $X = (S^1)^m \wedge G_m^{\wedge q}$ with $m>1$. It follows that there is an EHP sequence spectral sequence $$Z_{(2)} \otimes π_{n+1+i}^{A1} (S^{2n+2m+1} \wedge (G_m)^{\wedge 2q}) \Rightarrow Z_{(2)} \otimes π_{i}^{A1, s} (S^{m} \wedge (G_m)^{\wedge q}).$$

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arXiv2018

An étale realization which does not exist

For a global field, local field, or finite field $k$ with infinite Galois group, we show that there can not exist a functor from the Morel--Voevodsky $\mathbb{A}^1$-homotopy category of schemes over $k$ to a genuine Galois equivariant homotopy category satisfying a list of hypotheses one might expect from a genuine equivariant category and an étale realization functor. For example, these hypotheses are satisfied by genuine $\mathbb{Z}/2$-spaces and the $\mathbb{R}$-realization functor constructed by Morel--Voevodsky. This result does not contradict the existence of étale realization functors to (pro-)spaces, (pro-)spectra or complexes of modules with actions of the absolute Galois group when the endomorphisms of the unit is not enriched in a certain sense. It does restrict enrichments to representation rings of Galois groups.

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arXiv2018

A classical proof that the algebraic homotopy class of a rational function is the residue pairing

Cazanave has identified the algebraic homotopy class of a rational function of $1$ variable with an explicit nondegenerate symmetric bilinear form. Here we show that Hurwitz's proof of a classical result about real rational functions essentially gives an alternative proof of the stable part of Cazanave's result. We also explain how this result can be interpreted in terms of the residue pairing and that this interpretation relates the result to the signature theorem of Eisenbud, Khimshiashvili, and Levine, showing that Cazanave's result answers a question posed by Eisenbud for polynomial functions in $1$ variable. Finally, we announce results answering this question for functions in an arbitrary number of variables.

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arXiv2018

The Galois action and cohomology of a relative homology group of Fermat Curves

For an odd prime $p$ satisfying Vandiver's conjecture, we give explicit formulae for the action of the absolute Galois group $G_{\mathbb{Q}(ζ_p)}$ on the homology of the degree $p$ Fermat curve, building on work of Anderson. Further, we study the invariants and the first Galois cohomology group which are associated with obstructions to rational points on the Fermat curve.

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arXiv2017

Classification of problematic subgroups of U(n)

We classify p-toral subgroups of U(n) that can have non-contractible fixed points under the action of U(n) on the complex of partitions of complex n-space into mutually orthogonal subspaces.

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arXiv2016

Massey products in Galois cohomology via rational points

For $x$ an element of a field other than $0$ or $1$, we compute the order $n$ Massey products $$\langle (1-x)^{-1}, x^{-1}, \ldots, x^{-1}, (1-x)^{-1} \rangle$$ of $n-2$ factors of $x^{-1}$ and two factors of $(1-x)^{-1}$ by embedding $\mathbb{P}^1 - \{0,1,\infty\}$ into its Picard variety and constructing $\operatorname{Gal}(k^s/k)$ equivariant maps from $π_1$ applied to this embedding to unipotent matrix groups. This method produces obstructions to $π_1$-sections of $\mathbb{P}^1 - \{0,1,\infty\}$, partial computations of obstructions of Jordan Ellenberg, and also computes the Massey products $$\langle x^{-1} , (-x)^{-1}, \ldots, (-x)^{-1}, x^{-1} \rangle.$$

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arXiv2016

The simplicial suspension sequence in A^1-homotopy

We study a version of the James model for the loop space of a suspension in unstable ${\mathbb A}^1$-homotopy theory. We use this model to establish an analog of G.W. Whitehead's classical refinement of the Freudenthal suspension theorem in ${\mathbb A}^1$-homotopy theory: our result refines F. Morel's ${\mathbb A}^1$-simplicial suspension theorem. We then describe some $E_1$-differentials in the EHP sequence in ${\mathbb A}^1$-homotopy theory. These results are analogous to classical results of G.W. Whitehead's. Using these tools, we deduce some new results about unstable ${\mathbb A}^1$-homotopy sheaves of motivic spheres, including the counterpart of a classical rational non-vanishing result.

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arXiv2015

Galois action on the homology of Fermat curves

In his paper titled "Torsion points on Fermat Jacobians, roots of circular units and relative singular homology", Anderson determines the homology of the degree $n$ Fermat curve as a Galois module for the action of the absolute Galois group $G_{\mathbb{Q}(ζ_n)}$. In particular, when $n$ is an odd prime $p$, he shows that the action of $G_{\mathbb{Q}(ζ_p)}$ on a more powerful relative homology group factors through the Galois group of the splitting field of the polynomial $1-(1-x^p)^p$. If $p$ satisfies Vandiver's conjecture, we prove that the Galois group of this splitting field over $\mathbb{Q}(ζ_p)$ is an elementary abelian $p$-group of rank $(p+1)/2$. Using an explicit basis for this Galois group, we completely compute the relative homology, the homology, and the homology of an open subset of the degree $3$ Fermat curve as Galois modules. We then compute several Galois cohomology groups which arise in connection with obstructions to rational points.

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arXiv2015

Desuspensions of S^1 /\ P^1_Q-{0,1,infty}

We use the Galois action on $π_1^{\textrm{et}}(\mathbb{P}_{\overline{\mathbb{Q}}}^1 - \{0,1,\infty \})$ to show that the homotopy equivalence $S^1 \wedge (\mathbb{G}_{m,\mathbb{Q}} \vee \mathbb{G}_{m,\mathbb{Q}}) \cong S^1 \wedge (\mathbb{P}_{\mathbb{Q}}^1 - \{0,1,\infty \}) $ coming from purity does not desuspend to a map $\mathbb{G}_{m,\mathbb{Q}} \vee \mathbb{G}_{m,\mathbb{Q}} \to \mathbb{P}_{\mathbb{Q}}^1 - \{0,1,\infty \}$.

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arXiv2014

An Abel map to the compactified Picard scheme realizes Poincaré duality

For a smooth algebraic curve X over a field, applying H_1 to the Abel map X -> Pic (X/\partial X) to the Picard scheme of X modulo its boundary realizes the Poincaré duality isomorphism H_1(X, Z/ n) -> H^1(X/ \partial X, Z/n(1)) = H^1_c(X, Z/n(1)). We show the analogous statement for the Abel map X/\partial X -> Picbar (X/\partial X) to the compactified Picard, or Jacobian, scheme, namely this map realizes the Poincaré duality isomorphism H_1(X/ \partial X, Z/n) -> H^1(X, Z/n(1)). In particular, H_1 of this Abel map is an isomorphism. In proving this result, we prove some results about Picbar that are of independent interest. The singular curve X/\partial X has a unique singularity that is an ordinary fold point, and we describe the compactified Picard scheme of such a curve up to universal homeomorphism using a presentation scheme. We construct a Mayer-Vietoris sequence for certain push-outs of schemes, and an isomorphism of functors π_1^{ell} Pic^0(-) = H^1(-,Z_ell(1)).

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arXiv2014

Splitting Varieties for Triple Massey Products

We construct splitting varieties for triple Massey products. For a,b,c in F^* the triple Massey product < a,b,c> of the corresponding elements of H^1(F, mu_2) contains 0 if and only if there is x in F^* and y in F[\sqrt{a}, \sqrt{c}]^* such that b x^2 = N_{F[\sqrt{a}, \sqrt{c}]/F}(y), where N_{F[\sqrt{a}, \sqrt{c}]/F} denotes the norm, and F is a field of characteristic different from 2. These varieties satisfy the Hasse principle by a result of D.B. Lee and A.R. Wadsworth. This shows that triple Massey products for global fields of characteristic different from 2 always contain 0.

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arXiv2014

Fixed points of $p$-toral groups acting on partition complexes

We consider the action of $p$-toral subgroups of $U(n)$ on the unitary partition complex $\mathcal L_n$. We show that if $H\subseteq U(n)$ is $p$-toral and has noncontractible fixed points on $\mathcal L_n$, then the image of $H$ in the projective unitary group $U(n)/S^{1}$ is an elementary abelian $p$-group.

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arXiv2013

Cartier's first theorem for Witt vectors on Z_{>= 0}^n - 0

We show that the dual of the Witt vectors on Z_{\geq 0}^n - 0 as defined by Angeltveit, Gerhardt, Hill, and Lindenstrauss represent the functor taking a commutative formal group G to the maps of formal schemes Ahat^n -> G, and that the Witt vectors are self-dual for Q-algebras or when n=1.

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arXiv2013

2-Nilpotent Real Section Conjecture

We show a 2-nilpotent section conjecture over R: for a geometrically connected curve X over R such that each irreducible component of its normalization has R-points, pi_0(X(R)) is determined by the maximal 2-nilpotent quotient of the fundamental group with its Galois action, as the kernel of an obstruction of Jordan Ellenberg. This implies that for X smooth and proper, X(R)^{+/-} is determined by the maximal 2-nilpotent quotient of Gal(C(X)) with its Gal(R)-action, where X(R)^{+/-} denotes the set of real points equipped with a real tangent direction, showing a 2-nilpotent birational real section conjecture.

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arXiv2012

3-nilpotent obstructions to pi_1 sections for P^1_Q - {0,1,infty}

We study which rational points of the Jacobian of P^1_K -{0,1,infty} can be lifted to sections of geometrically 3 nilpotent quotients of etale pi_1 over the absolute Galois group. This is equivalent to evaluating certain triple Massey products of elements of H^1(G_K). For K=Q_p or R, we give a complete mod 2 calculation. This permits some mod 2 calculations for K = Q. These are computations of obstructions of Jordan Ellenberg.

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arXiv2011

n-Nilpotent Obstructions to pi_1 Sections of P^1-{0,1,infty} and Massey Products

Let pi be a pro-l completion of a free group, and let G be a profinite group acting continuously on pi. First suppose the action is given by a character. Then the boundary maps delta_n: H^1(G, pi/[pi]_n) -> H^2(G, [pi]_n/[pi]_{n+1}) are Massey products. When the action is more general, we partially compute these boundary maps. Via obstructions of Jordan Ellenberg, this implies that pi_1 sections of P^1_k-{0,1,infty} satisfy the condition that associated nth order Massey products in Galois cohomology vanish. For the pi_1 sections coming from rational points, these conditions imply that < (1-x)^{-1}, x^{-1}, x^{-1}, ..., x^{-1} > = 0 where x in H^1(Gal_k, Z_l(chi)) is the image of an element of k^* under the Kummer map.

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