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References for variations of Seifert-van Kampen'SSeifert–van Kampen's theorem: HNN extensions and "sensible" intersections

A basic consequence of the Seifert van-KampenSeifert–van Kampen theorem is the following.

Theorem: Consider a union of topological spaces $X,Y$$X$, $Y$ whose intersection $X\cap Y = Z$ is open connected and $\pi_1$-injective. Then for a basepoint $z_0 \in Z$ we have $$ \pi_1(X\cup Y, z_0) = \pi_1(X,z_0)*_{\pi_1(Z,z_0)}\pi_1(Y,z_0), $$ i.e. the fundamental group of the union is the amalgamated product.

I am looking for references for two variations of this result:

  1. What if we assume $X,Y$$X$, $Y$ are compact CW-complexes and $Z = X \cap Y$ is a $\pi_1$-injective subcomplex? The difference here is that $Z$ is no longer open but as a subcomplex it's still a sensible subspace.
  2. $X$ has two disjoint homeomorphic $\pi_1$-injective sensible subspaces $Z_1,Z_2 \subset X$ (e.g. points, circles) and the quotient space $Y = X/\sim$$Y = X/{\sim}$ is obtained by identifying $Z_1,Z_2$$Z_1$, $Z_2$. Then $\pi_1(Y)$ is isomorphic to an HNN extension of $\pi_1(X)$.

The type of application I'm thinking about is stuff along the lines of identifying two boundary components of a surface with boundary or identifying two different points in a space and then working out the resulting fundamental group.

I'm not happy with what is in Munkres, Hatcher, or Scott and Wall, in that there are no clear results I can just give to a grad student (maybe that's the point though.). Is anybody aware of other references that provide a good account of this or is it just folklore?

References for variations of Seifert-van Kampen'S theorem: HNN extensions and "sensible" intersections

A basic consequence of the Seifert van-Kampen theorem is the following.

Theorem: Consider a union of topological spaces $X,Y$ whose intersection $X\cap Y = Z$ is open connected and $\pi_1$-injective. Then for a basepoint $z_0 \in Z$ we have $$ \pi_1(X\cup Y, z_0) = \pi_1(X,z_0)*_{\pi_1(Z,z_0)}\pi_1(Y,z_0), $$ i.e. the fundamental group of the union is the amalgamated product.

I am looking for references for two variations of this result:

  1. What if we assume $X,Y$ are compact CW-complexes and $Z = X \cap Y$ is a $\pi_1$-injective subcomplex? The difference here is that $Z$ is no longer open but as a subcomplex it's still a sensible subspace.
  2. $X$ has two disjoint homeomorphic $\pi_1$-injective sensible subspaces $Z_1,Z_2 \subset X$ (e.g. points, circles) and the quotient space $Y = X/\sim$ is obtained by identifying $Z_1,Z_2$. Then $\pi_1(Y)$ is isomorphic an HNN extension of $\pi_1(X)$.

The type of application I'm thinking about is stuff along the lines of identifying two boundary components of a surface with boundary or identifying two different points in a space and then working out the resulting fundamental group.

I'm not happy with what is in Munkres, Hatcher, or Scott and Wall, in that there are no clear results I can just give to a grad student (maybe that's the point though.) Is anybody aware of other references that provide a good account of this or is it just folklore?

References for variations of Seifert–van Kampen's theorem: HNN extensions and "sensible" intersections

A basic consequence of the Seifert–van Kampen theorem is the following.

Theorem: Consider a union of topological spaces $X$, $Y$ whose intersection $X\cap Y = Z$ is open connected and $\pi_1$-injective. Then for a basepoint $z_0 \in Z$ we have $$ \pi_1(X\cup Y, z_0) = \pi_1(X,z_0)*_{\pi_1(Z,z_0)}\pi_1(Y,z_0), $$ i.e. the fundamental group of the union is the amalgamated product.

I am looking for references for two variations of this result:

  1. What if we assume $X$, $Y$ are compact CW-complexes and $Z = X \cap Y$ is a $\pi_1$-injective subcomplex? The difference here is that $Z$ is no longer open but as a subcomplex it's still a sensible subspace.
  2. $X$ has two disjoint homeomorphic $\pi_1$-injective sensible subspaces $Z_1,Z_2 \subset X$ (e.g. points, circles) and the quotient space $Y = X/{\sim}$ is obtained by identifying $Z_1$, $Z_2$. Then $\pi_1(Y)$ is isomorphic to an HNN extension of $\pi_1(X)$.

The type of application I'm thinking about is stuff along the lines of identifying two boundary components of a surface with boundary or identifying two different points in a space and then working out the resulting fundamental group.

I'm not happy with what is in Munkres, Hatcher, or Scott and Wall, in that there are no clear results I can just give to a grad student (maybe that's the point though). Is anybody aware of other references that provide a good account of this or is it just folklore?

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References for variations of Seifert-van Kampen'S theorem: HNN extensions and "sensible" intersections

A basic consequence of the Seifert van-Kampen theorem is the following.

Theorem: Consider a union of topological spaces $X,Y$ whose intersection $X\cap Y = Z$ is open connected and $\pi_1$-injective. Then for a basepoint $z_0 \in Z$ we have $$ \pi_1(X\cup Y, z_0) = \pi_1(X,z_0)*_{\pi_1(Z,z_0)}\pi_1(Y,z_0), $$ i.e. the fundamental group of the union is the amalgamated product.

I am looking for references for two variations of this result:

  1. What if we assume $X,Y$ are compact CW-complexes and $Z = X \cap Y$ is a $\pi_1$-injective subcomplex? The difference here is that $Z$ is no longer open but as a subcomplex it's still a sensible subspace.
  2. $X$ has two disjoint homeomorphic $\pi_1$-injective sensible subspaces $Z_1,Z_2 \subset X$ (e.g. points, circles) and the quotient space $Y = X/\sim$ is obtained by identifying $Z_1,Z_2$. Then $\pi_1(Y)$ is isomorphic an HNN extension of $\pi_1(X)$.

The type of application I'm thinking about is stuff along the lines of identifying two boundary components of a surface with boundary or identifying two different points in a space and then working out the resulting fundamental group.

I'm not happy with what is in Munkres, Hatcher, or Scott and Wall, in that there are no clear results I can just give to a grad student (maybe that's the point though.) Is anybody aware of other references that provide a good account of this or is it just folklore?