James reduced product construction

James reduced product is the free monoid generated by elements of in the category of topological spaces. We would like to find out its homology. Note that is obtained from by the following pushout diagram. Thus we see that is with attached. To compute the homology of inductively, we note that if we take (reduced) suspension of this diagram (which remains a pushout since being left adjoint preserves colimit), the vertical column splits since . Thus . Thus we have The last isomorphism follows from Kunneth and if is a cofibration (which implies that is the quotient of by ).

To compute the cohomology ring of , note that from its cellular structure we know that has exactly one at degree for every . To figure out the ring structure, let be a generator of the degree component of . We first compute for any . Note that this is a multiple of , say for some . Let be the attaching map, which induces . We will compute via relating the image of and under . Note that by Kunneth the cohomology ring of is , the exterior algebra with generators at degree . It is easy to see that the image of under is . To compute , note that the preimage of under is , and it is easy to see from this that . Thus Since and pairwise commute, we deduce that .

From knowing we show easily. This implies the (integral) cohomology of is the divided power algebra on -dimensional class.

Reference: https://en.wikipedia.org/wiki/James_reduced_product

It’s possible to include a diagram generated by quiver in your blog.

Click the meta menu on the editor toobar and add the following style code provided by quiver to your latex preamble:

% *** quiver ***
% A package for drawing commutative diagrams exported from https://q.uiver.app.
%
% This package is currently a wrapper around the `tikz-cd` package, importing necessary TikZ
% libraries, and defining new TikZ styles for curves of a fixed height and for shortening paths
% proportionally.
%
% Version: 1.6.0
% Authors:
% - varkor (https://github.com/varkor)
% - AndréC (https://tex.stackexchange.com/users/138900/andr%C3%A9c)
% - Andrew Stacey (https://tex.stackexchange.com/users/86/andrew-stacey)

\NeedsTeXFormat{LaTeX2e}
\ProvidesPackage{quiver}[2025/09/20 quiver]

% `tikz-cd` is necessary to draw commutative diagrams.
\RequirePackage{tikz-cd}
% `amssymb` is necessary for `\lrcorner` and `\ulcorner`.
\RequirePackage{amssymb}
% `calc` is necessary to draw curved arrows.
\usetikzlibrary{calc}
% `pathmorphing` is necessary to draw squiggly arrows.
\usetikzlibrary{decorations.pathmorphing}
% `spath3` is necessary to draw shortened edges.
\usetikzlibrary{spath3}

% A TikZ style for curved arrows of a fixed height, due to AndréC.
\tikzset{curve/.style={settings={#1},to path={(\tikztostart)
    .. controls ($(\tikztostart)!\pv{pos}!(\tikztotarget)!\pv{height}!270:(\tikztotarget)$)
    and ($(\tikztostart)!1-\pv{pos}!(\tikztotarget)!\pv{height}!270:(\tikztotarget)$)
    .. (\tikztotarget)\tikztonodes}},
    settings/.code={\tikzset{quiver/.cd,#1}
        \def\pv##1{\pgfkeysvalueof{/tikz/quiver/##1}}},
    quiver/.cd,pos/.initial=0.35,height/.initial=0}

% A TikZ style for shortening paths without the poor behaviour of `shorten <' and `shorten >'.
\tikzset{between/.style n args={2}{/tikz/execute at end to={
    \tikzset{spath/split at keep middle={current}{#1}{#2}}
}}}

% TikZ arrowhead/tail styles.
\tikzset{tail reversed/.code={\pgfsetarrowsstart{tikzcd to}}}
\tikzset{2tail/.code={\pgfsetarrowsstart{Implies[reversed]}}}
\tikzset{2tail reversed/.code={\pgfsetarrowsstart{Implies}}}
% TikZ arrow styles.
\tikzset{no body/.style={/tikz/dash pattern=on 0 off 1mm}}

%\endinput

Note that the last line \endinput is commented out, as the above code snippet is embedded directly in the preamble rather than being used as a standalone style file.

James reduced product construction

Comments

Add a comment