Steenrod coalgebras III. The fundamental group

2014/03/08 by Justin R Smith, Smith, Justin R.
Mathematics · #Homotopy and Cohomology in Algebraic Topology #Geometric and Algebraic Topology

paper · pdf · doi:10.48550/arxiv.1403.1973

Abstract

In this note, we extend earlier work by showing that if X and Y are delta-complexes (i.e. simplicial sets without degeneracy operators), a morphism g:N(X)→ N(Y) of Steenrod coalgebras (normalized chain-complexes equipped with extra structure) induces one of 2-skeleta g:X₂→ Y₂, inducing a homomorphism π₁(g):π₁(X)→π₁(Y) that is an isomorphism if g is an isomorphism. This implies a corresponding conclusion for a morphism g:C(X)→ C(Y) of Steenrod coalgebras on unnormalized chain-complexes of simplicial sets.

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