(Co)fibrations, suspensions, and loop spaces

 Seminar topic

Recall the exponential object $Z^Y$, which, in the category of topological spaces, is the set of all continuous functions $Y\to Z$. In general, the definition involves a commuting diagram and gives an isomorphism $\Hom(X\times Y,Z)\cong \Hom(X,Z^Y)$. The subspace $F(Y,Z)$ of $Z^Y$ consists of based functions $Y\to Z$.

Definition: Let $F,E,B,X$ be topological spaces. A map $i:F\to E$ is a cofibration if for every map $f:E\to X$ and every homotopy $h:F\times I\to X$, there exists a homotopy $\tilde h:E\times I\to X$ (extending $h$) making either of the equivalent diagrams below commute.

The horizontal maps on the left are the natural inclusion maps $x\mapsto (x,0)$ and the map on the right is the natural evaluation map $\varphi \mapsto \varphi(0)$. Similarly, a map $p:E\to B$ is a fibration if for every map $g:X\to E$ and every homotopy $h:X\times I\to B$, there exists a homotopy $\tilde h:X\times I\to E$ (lifting $h$) making either of the equivalent diagrams below commute.

The horizontal maps on the right are the natural evaluation maps and the map on the right is the natural inclusion map.

Instead of this terminology, often we say the pair $(F,E)$ has the homotopy extension property and the pair $(E,B)$ has the homotopy lifting property. Now, let let $(X,x)$ be a pointed topological space.

Definition: The (reduced) suspension $\Sigma X$ of $X$ is
\[
\Sigma X := X\times I/X\times \{0\} \cup X\times \{1\} \cup \{x\}\times I.
\] 
The unreduced suspension $SX$ of $X$ is
\[
S X := X\times I/X\times \{0\} \cup X\times \{1\}.
\]
The loop space $\Omega X$ of $X$ is
\[
\Omega X := F(S^1,X).
\]
Remark: If $X$ is well-pointed (the inclusion $i:\{x\}\hookrightarrow X$ is a cofibration), then the natural quotient map $SX\to \Sigma X$ is a homotopy equivalence. Moreover, there is an adjunction $F(\Sigma X,Y)\cong F(X,\Omega Y)$. In the fundamental group this gives the adjunction
\[
[\Sigma X,Y]\cong [X,\Omega Y],
\]
where $[A,B]$ is the set of based homotopy classes of maps $A\to B$.

References: May (A concise course in algebraic toplogy, Chapters 6, 7, 8), Aguilar, Gitler, and Prieto (Algebraic topology from a homotopical viewpoint, Chapter 2.10)

Ghost maps

 Seminar topic

Definition: Let $X$ be a topological space based at $x\in X$. Let $PX$ be the space of based paths of $X$, that is, maps $[0,1]\to X$ with $0\mapsto x$. Let $\Omega X\subset PX$ be the space of based loops of $X$, that is, maps $[0,1]\to X$ with $0,1\mapsto x$.

Note that $\Omega$ is a functor on the category of based topoloigcal spaces right-adjoint to the suspension functor $\Sigma$. Also observe there is a fibration
\[
\Omega X \to PX \tov{p} X,
\]
where $p$ is evaluation at $1\in [0,1]$. Since $PX$ is contractible, $H_n(PX)=0$ for $n\neq 0$, so $H_1(\Omega X) \cong H_2(X)$.

Definition: A spectrum $E$ is a sequence of based topological spaces $(E_n,x_n)$ and based homeomorphisms $\alpha_n:E_n \to \Omega E_{n+1}$. A map of spectra $f:E\to F$ is a sequence of based homeomorphisms $f_n:E_n\to F_n$ compatible with the based homeomorphisms of $E$ and $F$, that is, so that the diagram

commutes for all $n$.

Definition: Let $E,F$ be spectra. A map of spectra $f:E\to F$ is a ghost map if the induced map $\pi_nf:\pi_nX \to \pi_n Y$ on stable homotopy groups is the zero map.

Most commonly this term is used in spectra, but the idea of a ghost map may be generalized to other situations, where a map induces the zero map on homology, cohomology, or some similar functor.

References: Weibel (An introduction to homological algebra, Chapters 5.3, 10.9)