This is going to basically be a cross-post of a MSE question: http://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).

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).

This is going to basically be a cross-post of a MSE question: http://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?

This is going to basically be a cross-post of a MSE question: http://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).