# Good pair vs. cofibration

It can be shown that $i:A\hookrightarrow X$ is a closed cofibration if and only if there is a map $\varphi:X\to I=[0,1]$ and a homotopy $H:U\times I\to X$ on some neighborhood $U$ of $A$ such that

• $A=φ^{-1}(0)$ and $φ^{-1}([0,1))\subseteq U$
• $H(x,0)=x$, $H(a,t)=a$, and $H(x,1)\in A$ for all $x\in U, a\in A, t\in I$.

The last condition says that the neighborhood $U$ is deformable in $X$ to $A$. However, this seems to be different from $U$ deformation retracting to $A$ since the path $H(\{x\}×I)$ need not stay in $U$.

We can call $(X,A)$ a good pair (terminology from Hatcher's Algebraic Topology) if $A$ is a closed subspace of $X$ and some neighborhood $V$ deformation retracts to $A$.

I'd be interested to see an example of a closed cofibration which is not a good pair. Such an example would feature a neighborhood $U$ of $A$ whose inclusion in $X$ is homotopic to a retraction onto $A$ but for no neighborhood such a homotopy would stay within that set. At first I thought you could simply replace $U$ be the larger set $V=H[U\times I]$ and for each point $y\in V$ let the path start at $t_y$ the smallest point in time such that some $y=H(x,t_y)$ for some $x$ in $U$, and let $y$ be attached to $x$ on its path towards $A$, but this doesn't always give a well-defined map. In other words, the deformable neighborhood does not always give rise to a neighborhood which deformation retracts to $A$.

Of course if someone comes up with a good pair which is not cofibered, that would be nice, too.

• I think that you are missing assumption about closeness of $A$. If so taking your assumptions about closed cofibration one may call pair $(A,X)$ neighbourhood deformation retract. Then this answer and comment bellow it may be useful in further research. – Stephen Dedalus Nov 1 '14 at 18:59
• @StephenDedalus I don't understand the comment. It seems like this is not really addressed in May or Hatcher, and I don't see anything helpful at the linked answer or in the blog post linked to therein. (That blog post says something about U deformation retracting to A, but just as in May's book that's not really part of the statement.) Arne Strom's papers Notes on Cofibrations I and II say more clearly that U is deformable to A in X rather than saying that U deformation retracts to A. – Dan Ramras Feb 29 '16 at 4:56
• Practically speaking, the important thing is to know that a cofibration induces a LES on homology (involving the homologies of A, X, and X/A). This can be seen by noting that the the inclusion of CA into the mapping cone Ci is a cofibration (it's obtained from i by a pushout) and $Ci\rightarrow Ci/CA = X/A$ is a homotopy equivalence (Hatcher 0.17). If Mi is the mapping cylinder, then (Mi, A) is good and gives the LES. More specifically, one sees in this manner that the natural map $H_*(X,A) \rightarrow H_* (X/A, A/A)$ is an isomorphism for any cofibration. – Dan Ramras Feb 29 '16 at 4:59
• I just noticed that my previous comment is more or less what May says in Section 14.2 of his book; he's working with an arbitrary generalized homology theory. – Dan Ramras Feb 29 '16 at 5:27