\(
\def\CC{{\mathbb C}}
\def\RR{{\mathbb R}}
\def\NN{{\mathbb N}}
\def\ZZ{{\mathbb Z}}
\def\QQ{{\mathbb Q}}
\def\TT{{\mathbb T}}
\def\CF{{\operatorname{CF}^\bullet}}
\def\HF{{\operatorname{HF}^\bullet}}
\def\SH{{\operatorname{SH}^\bullet}}
\def\ot{{\leftarrow}}
\def\st{\;:\;}
\def\Fuk{{\operatorname{Fuk}}}
\def\emprod{m}
\def\cone{\operatorname{Cone}}
\def\Flux{\operatorname{Flux}}
\def\li{i}
\def\ev{\operatorname{ev}}
\def\id{\operatorname{id}}
\def\grad{\operatorname{grad}}
\def\ind{\operatorname{ind}}
\def\weight{\operatorname{wt}}
\def\Sym{\operatorname{Sym}}
\def\HeF{\widehat{CHF}^\bullet}
\def\HHeF{\widehat{HHF}^\bullet}
\def\Spinc{\operatorname{Spin}^c}
\def\min{\operatorname{min}}
\def\div{\operatorname{div}}
\def\SH{{\operatorname{SH}^\bullet}}
\def\CF{{\operatorname{CF}^\bullet}}
\def\Tw{{\operatorname{Tw}}}
\def\Log{{\operatorname{Log}}}
\def\TropB{{\operatorname{TropB}}}
\def\wt{{\operatorname{wt}}}
\def\Span{{\operatorname{span}}}
\def\Crit{\operatorname{Crit}}
\def\CritVal{\operatorname{CritVal}}
\def\FS{\operatorname{FS}}
\def\Sing{\operatorname{Sing}}
\def\Coh{\operatorname{Coh}}
\def\Vect{\operatorname{Vect}}
\def\into{\hookrightarrow}
\def\tensor{\otimes}
\def\CP{\mathbb{CP}}
\def\eps{\varepsilon}
\)
Symplectic snippets
some properties of topological mapping cone
In the homotopy category of pointed spaces, the mapping cone takes a special meaning. Let \(f: (X,x)\to (Y, y)\) be a pointed map. From this we can form \((Z, z):=(\cone(f), y)\) which is again a pointed space. We now address some relations between spaces \((X, x), (Y, y)\) and \((Z, z)\).
- Observe that the composition \(i\circ f: (X, x)\to (Z, x))\) is homotopic to the constant map. In the homotopy category of pointed spaces, we can therefore write \(i\circ f \sim 0\), where \(0: (X, x)\to (Z, x)\) is the map factoring through a point.
- We can additionally look at the cone
\[(Y, y)\xrightarrow{i}(Z, z).\]
This second cone can be rewritten in terms of the data \(f, X\) and \(Y\) as
\[ (Y\times J)\cup ((X\times I)\cup Y / \sim\]
where the relations are
\[(x_1, 0)\sim (x_2, 0) , (x, 1)\sim f(x)) , (y_1, 0)\sim (y_2, 0), (y, 1)\sim y.\]
This is homotopic to the suspension \(\Sigma X\) allowing us to write the ``long exact sequence''
\[(X, x)\to (Y, y)\to (Z, z)\to (\Sigma X, x)\to (\Sigma Y, y)\to \cdots\]
in the homotopy category of pointed spaces.