Template:Arone&Krushkal2021

G. Arone, V. Krushkal, Embedding obstructions in $\R^d$$\newcommand{\Z}{\mathbb{Z}} \newcommand{\R}{\mathbb{R}} \newcommand{\C}{\mathbb{C}} \newcommand{\N}{\mathbb{N}} \newcommand{\Q}{\mathbb{Q}} \newcommand{\F}{\mathbb{F}} \newcommand{\bZ}{\mathbb{Z}} \newcommand{\bR}{\mathbb{R}} \newcommand{\bC}{\mathbb{C}} \newcommand{\bH}{\mathbb{H}} \newcommand{\bQ}{\mathbb{Q}} \newcommand{\bF}{\mathbb{F}} \newcommand{\bN}{\mathbb{N}} \DeclareMathOperator\id{id} % identity map \DeclareMathOperator\Sq{Sq} % Steenrod squares \DeclareMathOperator\Homeo{Homeo} % group of homeomorphisms of a topoloical space \DeclareMathOperator\Diff{Diff} % group of diffeomorphisms of a smooth manifold \DeclareMathOperator\SDiff{SDiff} % diffeomorphism under some constraint \DeclareMathOperator\Hom{Hom} % homomrphism group \DeclareMathOperator\End{End} % endomorphism group \DeclareMathOperator\Aut{Aut} % automorphism group \DeclareMathOperator\Inn{Inn} % inner automorphisms \DeclareMathOperator\Out{Out} % outer automorphism group \DeclareMathOperator\vol{vol} % volume \newcommand{\GL}{\text{GL}} % general linear group \newcommand{\SL}{\text{SL}} % special linear group \newcommand{\SO}{\text{SO}} % special orthogonal group \newcommand{\O}{\text{O}} % orthogonal group \newcommand{\SU}{\text{SU}} % special unitary group \newcommand{\Spin}{\text{Spin}} % Spin group \newcommand{\RP}{\Rr\mathrm P} % real projective space \newcommand{\CP}{\Cc\mathrm P} % complex projective space \newcommand{\HP}{\Hh\mathrm P} % quaternionic projective space \newcommand{\Top}{\mathrm{Top}} % topological category \newcommand{\PL}{\mathrm{PL}} % piecewise linear category \newcommand{\Cat}{\mathrm{Cat}} % any category \newcommand{\KS}{\text{KS}} % Kirby-Siebenmann class \newcommand{\Hud}{\text{Hud}} % Hudson torus \newcommand{\Ker}{\text{Ker}} % Kernel \newcommand{\underbar}{\underline} %Classifying Spaces for Families of Subgroups \newcommand{\textup}{\text} \newcommand{\sp}{^}\R^d$ from the Goodwillie-Weiss calculus and Whitney disks, arxiv preprint.

1 Comment

• It is simple and known since [Wu1965] that if a space $K$$K$ embeds into $\R^d$$\R^d$, then for any $r$$r$ there exists a $\Sigma_r$$\Sigma_r$-equivariant map $C(K,r)\to C(\R^d,r)$$C(K,r)\to C(\R^d,r)$, see also survey [Skopenkov2006, end of $\S$$\S$5]. It would be nice to know if (for any $r$$r$, not only for $r=2,3$$r=2,3$) $O_r$$O_r$ is the primary obstruction (in the sense of classical obstruction theory) to the existence of a $\Sigma_r$$\Sigma_r$-equivariant map $C_s(K,r)\to C(\R^4,r)$$C_s(K,r)\to C(\R^4,r)$ agreeing with the given $\Sigma_j$$\Sigma_j$-equivariant maps $C_s(K,j) \to C(\R^4,j)$$C_s(K,j) \to C(\R^4,j)$, $j=1,2,...,r-1$$j=1,2,...,r-1$. Cf. Proposition 2.11. This would clarify the relation of Goodwillie-Weiss calculus (in this particular situation) to classical algebraic topology.

• It would be nice to mention that analogous (but simpler) obstruction to the existence of a map without $r$$r$-fold points (more precisely, of an almost $r$$r$-embedding) is known. See e.g. the survey [Skopenkov2016, end of $\S$$\S$1.3]. This analogy suggests that it would be nice to know if the Arone-Krushkal obstructions

- are trivial for $r$$r$ not a prime power (for the analogous result on obstruction to $r$$r$-almost embeddability see the survey [Skopenkov2016, Theorem 3.3 (\"Ozaydin)]);

- have finite order for any $r$$r$ (for the analogous obstruction $v(\Sigma_r)$$v(\Sigma_r)$ to $r$$r$-almost embeddability we have $\frac{r!}{p^{\alpha_p}}v(\Sigma_r)=0$$\frac{r!}{p^{\alpha_p}}v(\Sigma_r)=0$, see the survey [Skopenkov2016, $\S$$\S$3.2]);

- are sufficient to almost-embeddability (for the analogous results on $r$$r$-almost embeddability see the survey [Skopenkov2016, Theorems 3.1 (Mabillard-Wagner) and 3.5]).

• For the well-known geometric definition of the Milnor number $\mu$$\mu$ see e.g. [Skopenkov2012, $\S$$\S$4.6, $\S$$\S$4.7]. Definition 3.4 is analogous to the following equivalent definition, which is presumably due to Matsumoto (see paper, page 2, the second paragraph after theorem 1, citation of [37] for the triple linking number case, i.e., n=1). Assume that $a_1,a_2,a_3\subset S^3$$a_1,a_2,a_3\subset S^3$ are pairwise disjoint oriented knots such that $\mbox{lk}(a_i,a_j)=0$$\mbox{lk}(a_i,a_j)=0$. Let $M_i\subset D^4$$M_i\subset D^4$, $i=1,2,3$$i=1,2,3$, be general position oriented singular disks spanned by the knots $a_i=\partial M_i$$a_i=\partial M_i$. Since $\mbox{lk}(a_i,a_j)=0$$\mbox{lk}(a_i,a_j)=0$, we have $M_i\cdot M_j=0$$M_i\cdot M_j=0$. So we can split the points from $M_i\cap M_j$$M_i\cap M_j$ into pairs of points having opposite signs. Take disjoint union $W_{ij}=W_{(i,j)}$$W_{ij}=W_{(i,j)}$ of the naturally oriented Whitney disks corresponding to this splitting. (The orientation of $W_{ij}$$W_{ij}$ is induced from the orientation on their boundary circles which are oriented from the negative intersection point to the positive intersection point along $M_i$$M_i$, and vice versa along $M_j$$M_j$.) Then
  $\displaystyle \mu(a_1,a_2,a_3) = W_{12}\cdot M_3+W_{23}\cdot M_1+W_{31}\cdot M_2.$ $$\displaystyle \mu(a_1,a_2,a_3) = W_{12}\cdot M_3+W_{23}\cdot M_1+W_{31}\cdot M_2.$$
  In this definition singular disks could not be replaced by Seifert surfaces because Borromean rings in $S^3$$S^3$ bound pairwise disjoint Seifert surfaces in $D^4$$D^4$.

• The usual name for transformations analogous to `stabilization' of Definition 3.7 (resulting in adding a coboundary to a cocycle) is `(generalized) van Kampen finger move'. For such transformations see Figure 1 and surveys [Skopenkov2006, $\S$$\S$4, $\S$$\S$8], [Skopenkov2016, end of $\S$$\S$3.3], [Skopenkov2018i, $\S$$\S$1.5.3].

• In $\S$$\S$8.4, 2nd paragraph, instead of `PL embeddability is established' there should be `PL almost embeddability is established'.

2 References

• [Skopenkov2006] A. Skopenkov, Embedding and knotting of manifolds in Euclidean spaces, in: Surveys in Contemporary Mathematics, Ed. N. Young and Y. Choi, London Math. Soc. Lect. Notes, 347 (2008) 248-342. Available at the arXiv:0604045.
• [Skopenkov2012] A. Skopenkov, Algebraic Topology From Algorithmic Standpoint, draft of a book (partly in Russian)
• [Skopenkov2016] A. Skopenkov, A user's guide to the topological Tverberg Conjecture, Russian Math. Surveys, 73:2 (2018), 323--353. Full updated version: arXiv:1605.05141.
• [Skopenkov2018i] A. Skopenkov, Invariants of graph drawings in the plane, Arnold Math. J., 6 (2020) 21-55. Full updated version: arXiv:1805.10237
• [Wu1965] W. T. Wu, A Theory of Embedding, Immersion and Isotopy of Polytopes in an Euclidean Space, Peking: Science Press, 1965.