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)