Homotopy similarity of maps. Maps of the circle
We describe the relation of $r$-similarity and finite-order invariants on the homotopy set $[S^1,Y]=π_1(Y)$.
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arXiv subjects
S. S. Podkorytov
Publications and source records attributed to S. S. Podkorytov.
Homotopy similarity of maps. Strong similarity
Given pointed cellular spaces $X$ and $Y$, $X$ compact, and an integer $r\ge0$, we define a relation $\overset r\approx$ on $[X,Y]$ and argue for the conjecture that it always coincides with the $r$-similarity $\overset r\sim$.
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Homotopy similarity of maps. Compositions
We describe the behaviour of the homotopy similarity relations and finite-order invariants under the function $[X,Y]\to[X,Z]$ induced by a map $Y\to Z$ strongly $r$-similar to the constant map.
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Homotopy similarity of maps
Given based cellular spaces X and Y, X compact, we define a sequence of increasingly fine equivalences on the based-homotopy set [X,Y].
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On the algebra of the Möbius crown
A commutative algebra over a field gives rise to a representation of the category of finite sets and surjective maps. We consider the restriction of this representation to the subcategory of sets of cardinality at most $r$. For each $r$, we present two non-isomorphic algebras that give rise to isomorphic representations of this subcategory.
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Straight homotopy invariants
Let $X$ and $Y$ be spaces and $M$ be an abelian group. A homotopy invariant $f\colon [X,Y]\to M$ is called straight if there exists a homomorphism $F\colon L(X,Y)\to M$ such that $f([a])=F(\langle a\rangle)$ for all $a\in C(X,Y)$. Here $\langle a\rangle\colon\langle X\rangle\to\langle Y\rangle$ is the homomorphism induced by $a$ between the abelian groups freely generated by $X$ and $Y$ and $L(X,Y)$ is a certain group of `admissible' homomorphisms. We show that all straight invariants can be expressed through a `universal' straight invariant of homological nature.
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Steenrod operations and the diagonal morphism
We show how to find the Steenrod operations in H^*(X) (the coefficients in F_p) given the diagonal morphism d_#:S_*(X)->S_*(X^p) and the action of the cyclic group C_p on S_*(X^p). Our construction needs no other data such as Eilenberg-Zilber morphisms.
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Commutative algebras and representations of the category of finite sets
We prove that two finite-dimensional commutative algebras over an algebraically closed field are isomorphic if and only if they give rise to isomorphic representations of the category of finite sets and surjective maps.
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Order of a homotopy invariant in the stable case
Let X and Y be CW-complexes, U be an abelian group, and f:[X,Y]->U be a map (a homotopy invariant). We say that f has order at most r if the characteristic function of the r'th Cartesian power of the graph of a continuous map a:X->Y Z-linearly determines f([a]). Suppose that the CW-complex X is finite and we are in the stable case: dim X<2n-1 and Y is (n-1)-connected. We prove that then the order of f equals its degree with respect to the Curtis filtration of the group [X,Y].
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An iterated sum formula for a spheroid's homotopy class modulo 2-torsion
Let $X$ be a simply connected pointed space with finitely generated homotopy groups. Let $Π_n(X)$ denote the set of all continuous maps $a:I^n\to X$ taking $\partial I^n$ to the basepoint. For $a\inΠ_n(X)$, let $[a]\inπ_n(X)$ be its homotopy class. For an open set $E\subset I^n$, let $Π(E,X)$ be the set of all continuous maps $a:E\to X$ taking $E\cap\partial I^n$ to the basepoint. For a cover $Γ$ of $I^n$, let $Γ(r)$ be the set of all unions of at most $r$ elements of $Γ$. Put $r=(n-1)!$. We prove that for any finite open cover $Γ$ of $I^n$ there exist maps $f_E:Π(E,X)\toπ_n(X)\otimes Z[1/2]$, $E\inΓ(r)$, such that $$ [a]\otimes1=\sum_{E\inΓ(r)} f_E(a|_E) $$ for all $a\inΠ_n(X)$.
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