Stems and Spectral Sequences

TL;DR

This paper introduces n-stems as a new categorical framework for understanding higher-order homotopy groups and spectral sequences, providing algebraic models that facilitate computations in algebraic topology.

Contribution

It defines the category Pstem[n] of n-stems and establishes a functor from spaces to this category, linking n-stems to spectral sequences and homotopy theory.

Findings

Associates (n+1)-truncated spectral sequences to simplicial n-stems.

Shows the spectral sequence for Postnikov n-stems is a truncation of the usual homotopy spectral sequence.

Provides algebraic models for low-degree n-stems to aid computations.

Abstract

We introduce the category Pstem[n] of n-stems, with a functor P[n] from spaces to Pstem[n]. This can be thought of as the n-th order homotopy groups of a space. We show how to associate to each simplicial n-stem Q an (n+1)-truncated spectral sequence. Moreover, if Q=P[n]X is the Postnikov n-stem of a simplicial space X, the truncated spectral sequence for Q is the truncation of the usual homotopy spectral sequence of X. Similar results are also proven for cosimplicial n-stems. They are helpful for computations, since n-stems in low degrees have good algebraic models.