Mayer-Vietoris implies Excision

Question

This is going to basically be a cross-post of a MSE question: https://math.stackexchange.com/questions/147146/mayer-vietoris-implies-excision. I suspect that the answer to this question will turn out to be fairly straightforward, but (a) I can't figure it out myself, (b) it's not in any of the obvious references, (c) it hasn't gotten any answers on MSE, and (d) the question honestly is relevant to my research.

Let $H_n$ be a sequence of functors from the category of locally compact Hausdorff spaces to the category of abelian groups which satisfy all of the usual Eilenberg-Steenrod axioms for a generalized homology theory, except the excision axiom is replaced by a Mayer-Vietoris axiom: whenever $X$ is the union of two closed subspaces $A$ and $B$ there is a (functorial) exact sequence

$$\to H_n(A \cap B) \to H_n(A) \oplus H_n(B) \to H_n(X) \to H_{n-1}(A \cap B) \to$$

According to the slogan "Mayer-Vietoris is equivalent to excision", one ought to be able to prove an excision theorem for $H_n$. This probably takes the form of a long exact sequence

$$\to H_n(A) \to H_n(X) \to H_n(X - A) \to H_{n-1}(A) \to$$

where $A$ is a closed subset of $X$, though if this isn't the right formulation I welcome corrections. I know how to go the other direction (excision implies Mayer-Vietoris), but this direction eludes me. Can anyone help?

EDIT: I guess I was a bit sloppy about my use of the term "Eilenberg-Steenrod Axioms" which require that $H_n$ be a functor of pairs. So let me be more precise. We'll work in the category of pointed locally compact (second countable if desired) Hausdorff spaces where the morphisms between $X$ and $Y$ are basepoint preserving continuous maps from the one point compactification of $X$ to the one point compactification of $Y$. Define a homotopy of maps between $X$ and $Y$ to be a morphism between $X \times [0,1]$ and $Y$.

I will declare that a sequence of covariant functors $H_n$ from this category to the category of abelian groups defines a generalized homology theory if:

• Each $H_n$ is a homotopy functor

• There is a functorial long exact sequence of the form above associated to any closed subset $A \subseteq X$.

Under these circumstances there is a Mayer-Vietoris sequence associated to any decomposition of $X$ into closed subspaces. A precise statement of my question is: if we replace the second axiom with a Mayer-Vietoris axiom, can we reconstruct the long exact sequence associated to a single closed subset?

However, I am flexible. In the setting that motivates this question, I really do have a notion of homology of pairs, so I would really be happy if somebody could deduce any version of the excision theorem from the existence of a Mayer-Vietoris sequence (perhaps with some extra conditions on the boundary map if it helps).

Answer 1

I must admit, I don't know whether you can just replace excision by Mayer-Vietoris in the usual Eilenberg-Steenrod axioms. But you can do something slightly different:

While the Eilenberg-Steenrod axioms speak of homology of pairs of spaces, you can define a homology theory on spaces also as a sequence of homotopy invariant functors $h_n$ fulfilling a Mayer-Vietoris sequence (as Steven pointed out, you have just some functorial map $h_n(X) \to h_{n-1}(A\cap B)$, which replace as a datum the boundary maps of the long exact sequence of a pair). Then you define the homology of a pair $(X,A)$ as $h_n(X,A) := \ker(h_n(C_AX) \to h_n(pt))$. Here, $C_A X$ stands for the mapping cone of the inclusion $A\hookrightarrow X$. The Mayer-Vietoris sequence (based on the covering $CA \cup X = C_AX$) implies the long exact sequence of the pair. Excision (i.e. $h_n(X,A) \cong h_n(X-B, A-B)$ for $\overline{B} \subset A^\circ$) is a homotopy fact in the sense that the inclusion $C_{A-B}(X-B) \to C_A X$ is a homotopy equivalence. I must admit, I didn't check the last point in detail, but it should be at least true if $B\to A$ is a closed cofibration and in general you might replace the inclusion via a cylinder construction by a closed cofibration.

Thus, you get a different axiomatization of a homology theory using the Mayer-Vietoris sequence [There is also a third axiomatization of a homology theory, using pointed spaces as input and having a sequence for the cone and a suspension isomorphism as axioms]. Of course, if you have already a functor defined on pairs of spaces, you have to check that something like $h_n(X,A) = h_n(C_AX, pt)$ holds.

I want to comment, I know this "absolute" axiomatization primarly from bordism theory, where you can show Mayer-Vietoris quite easily.

Answer 2

I'd have taken "excision" to mean that the map $H_n(A,A\cap B)\rightarrow H_n(X,B)$ is an isomorphism.

I'd have taken "Mayer-Vietoris" to mean not just the exactness of some sequence $$\cdots\rightarrow H_n(A\cap B)\rightarrow H_n(A)\oplus H_n(B)\rightarrow H_n(X)\rightarrow H_{n-1}(A\cap B)\rightarrow\cdots$$ but the exactness of the particular sequence you get when the map $H_n(X)\rightarrow H_{n-1}(A\cap B)$ is defined by composing first the natural map from $H_n(X)\rightarrow H_n(X,B)$, then the inverse of the excision isomorphism, and then the natural map $H_{n}(A,A\cap B)\rightarrow H_{n-1}(A\cap B)$.

Given that, Mayer-Vietoris implies excision in the sense that the very statement of the Mayer-Vietoris property makes no sense in the absence of the excision isomorphism.

This does raise the question of whether a "Mayer-Vietoris type exact sequence" could occur in the absence of excision, with some other functorial map $H_n(X)\rightarrow H_{n-1}(A\cap B)$ replacing the one defined via excision. I'm not sure whether we'd still want to call this a Mayer-Vietoris sequence.