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Register a new userOn the Pontryagin-Steenrod-Wu theorem
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This paper is on homotopy classification of maps of (n+1)-dimensional manifolds into the n-dimensional sphere. For a continuous map f of an (n+1)-manifold into the n-sphere define the degree deg f to be the class dual to f^*[S^n], where [S^n] is the fundamental class. We present a short and direct proof of the following specific case of the Pontryagin-Steenrod-Wu theorem: Theorem. Let M be a connected orientable closed smooth (n+1)-manifold, n>2. Then the map deg:pi^n(M)to H_1(M;Z) is 1-to-1 (i.e., bijective), if the product w_2(M) x r_2 H_2(M;Z) is nonzero, where r_2 is the mod2 reduction; 2-to-1 (i.e., each element of H_1(M;Z) has exactly 2 preimages) - otherwise. The proof is based on the Pontryagin-Thom construction and a geometric definition of the Stiefel-Whitney classes w_2(M).
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We compute the $C_p$-equivariant dual Steenrod algebras associated to the constant Mackey functors $underline{mathbb{F}}_p$ and $underline{mathbb{Z}}_{(p)}$, as $underline{mathbb{Z}}_{(p)}$-modules. The $C_p$-spectrum $underline{mathbb{F}}_p otimes underline{mathbb{F}}_p$ is not a direct sum of $RO(C_p)$-graded suspensions of $underline{mathbb{F}}_p$ when $p$ is odd, in contrast with the classical and $C_2$-equivariant dual Steenrod algebras.
Conway and Gordon proved that for every spatial complete graph on six vertices, the sum of the linking numbers over all of the constituent two-component links is odd, and Kazakov and Korablev proved that for every spatial complete graph with arbitrary number of vertices greater than six, the sum of the linking numbers over all of the constituent two-component Hamiltonian links is even. In this paper, we show that for every spatial complete graph whose number of vertices is greater than six, the sum of the square of the linking numbers over all of the two-component Hamiltonian links is determined explicitly in terms of the sum over all of the triangle-triangle constituent links. As an application, we show that if the number of vertices is sufficiently large then every spatial complete graph contains a two-component Hamiltonian link whose absolute value of the linking number is arbitrary large. Some applications to rectilinear spatial complete graphs are also given.
Steenrod homotopy theory is a framework for doing algebraic topology on general spaces in terms of algebraic topology of polyhedra; from another viewpoint, it studies the topology of the lim^1 functor (for inverse sequences of groups). This paper is primarily concerned with the case of compacta, in which Steenrod homotopy coincides with strong shape. We attempt to simplify foundations of the theory and to clarify and improve some of its major results. Using geometric tools such as Milnors telescope compactification, comanifolds (=mock bundles) and the Pontryagin-Thom Construction, we obtain new simple proofs of results by Barratt-Milnor; Cathey; Dydak-Segal; Eda-Kawamura; Edwards-Geoghegan; Fox; Geoghegan-Krasinkiewicz; Jussila; Krasinkiewicz-Minc; Mardesic; Mittag-Leffler/Bourbaki; and of three unpublished results by Shchepin. An error in Lisitsas proof of the Hurewicz theorem in Steenrod homotopy is corrected. It is shown that over compacta, R.H.Foxs overlayings are same as I.M.James uniform covering maps. Other results include: - A morphism between inverse sequences of countable (possibly non-abelian) groups that induces isomorphisms on inverse and derived limits is invertible in the pro-category. This implies the Whitehead theorem in Steenrod homotopy, thereby answering two questions of A.Koyama. - If X is an LC_{n-1} compactum, n>0, its n-dimensional Steenrod homotopy classes are representable by maps S^nto X, provided that X is simply connected. The assumption of simply-connectedness cannot be dropped by a well-known example of Dydak and Zdravkovska. - A connected compactum is Steenrod connected (=pointed 1-movable) iff every its uniform covering space has countably many uniform connected components.
We establish a form of the h-principle for the existence of foliations quasi-complementary to a given one; the same methods also provide a proof of the classical Mather-Thurston theorem.
In 2019 P. Patak and M. Tancer obtained the following higher-dimensional generalization of the Heawood inequality on embeddings of graphs into surfaces. We expose this result in a short well-structured way accessible to non-specialists in the field. Let $Delta_n^k$ be the union of $k$-dimensional faces of the $n$-dimensional simplex. Theorem. (a) If $Delta_n^k$ PL embeds into the connected sum of $g$ copies of the Cartesian product $S^ktimes S^k$ of two $k$-dimensional spheres, then $ggedfrac{n-2k}{k+2}$. (b) If $Delta_n^k$ PL embeds into a closed $(k-1)$-connected PL $2k$-manifold $M$, then $(-1)^k(chi(M)-2)gedfrac{n-2k}{k+1}$.
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