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A typical example arising in algebraic geometry is the following.

Take a scheme $X$ of finite type over $\mathbb{C}$. Then there are (at least) two concepts of fundamental group for $X$: the topological fundamental group $\pi_1^{top}(X)$, i.e. the fundamental group of the underlying topological space $X(\mathbb{C})$ with the analytic topology, and the étale fundamental group $\pi^{et}(X)$, which is a purely algebraic object (it is an inverse limit of finite automorphism groups).

By the so-called Riemann Existence Theorem, the finite covering of $X(\mathbb{C})$, that a priori are only analytic spaces, have actually a scheme structure. This implies that $\pi_1^{et}(X)$ is precisely the profinite completion of $\pi_1^{top}(X)$.

Therefore there is a group homomorphism $$\eta \colon \pi_1^{top}(X) \longrightarrow \pi_1^{et}(X),$$ which is injective precisely when $\pi_1^{top}(X)$ is residually finite.

For instance, if for some reason we know that $\pi_1^{top}(X)$ is residually finite and we are able to show that $\pi_1^{et}(X)=0,$ then we can conclude that $\pi_1^{top}(X)=0,$ too.

J. P. Serre asked if there exist smooth, complex varieties such that $\pi_1^{top}(X)$ is not residually finite (and hence $\eta$ is not injective). A consequence of a theorem of Malcev is that if $\pi_1^{top}(X)$ has a faithful linear representation, then it is residually finite. Hence if $X$ is a variety answering affirmatively Serre's question, its topological fundamental group must have no faithful linear representation. Such varieties actually exist, and the first examples were constructed by D. Toledo, see

Projective varieties with non-residually finite fundamental group, Publications Mathématiques de l'Institut des Hautes Études Scientifiques Volume 77, Issue 1 (1993), 103-119.

In Toledo's examples, $\ker \eta$ is free group of infinite rank. Later, F. Catanese and J. Kollar, and independently M. Nori, were able to find examples where $\ker \eta$ is a finite cyclic group. Such examples are described in

Classification of irregular varieties, Lecture Notes in Mathematics 1515, Springer 1992.

A typical example arising in algebraic geometry is the following.

Take a scheme $X$ of finite type over $\mathbb{C}$. Then there are (at least) two concepts of fundamental group for $X$: the topological fundamental group $\pi_1^{top}(X)$, i.e. the fundamental group of the underlying topological space $X(\mathbb{C})$ with the analytic topology, and the étale fundamental group $\pi^{et}(X)$, which is a purely algebraic object (it is an inverse limit of finite automorphism groups).

By the so-called generalized Riemann Existence Theorem, whose proof is due to Grauert and Remmert, the finite coverings of $X(\mathbb{C})$, that a priori are only analytic spaces, have actually a scheme structure. This implies that $\pi_1^{et}(X)$ is precisely the profinite completion of $\pi_1^{top}(X)$.

Therefore there is a group homomorphism $$\eta \colon \pi_1^{top}(X) \longrightarrow \pi_1^{et}(X),$$ which is injective precisely when $\pi_1^{top}(X)$ is residually finite.

For instance, if for some reason we know that $\pi_1^{top}(X)$ is residually finite and we are able to show that $\pi_1^{et}(X)=0,$ then we can conclude that $\pi_1^{top}(X)=0,$ too.

J. P. Serre asked if there exist smooth, complex varieties such that $\pi_1^{top}(X)$ is not residually finite (and hence $\eta$ is not injective). A consequence of a theorem of Malcev is that if $\pi_1^{top}(X)$ has a faithful linear representation, then it is residually finite. Hence if $X$ is a variety answering affirmatively Serre's question, its topological fundamental group must have no faithful linear representation. Such varieties actually exist, and the first examples were constructed by D. Toledo, see

Projective varieties with non-residually finite fundamental group, Publications Mathématiques de l'Institut des Hautes Études Scientifiques Volume 77, Issue 1 (1993), 103-119.

In Toledo's examples, $\ker \eta$ is free group of infinite rank. Later, F. Catanese and J. Kollar, and independently M. Nori, were able to find examples where $\ker \eta$ is a finite cyclic group. Such examples are described in

Classification of irregular varieties, Lecture Notes in Mathematics 1515, Springer 1992.

A typical example arising in algebraic geometry is the following.

Take a scheme $X$ of finite type over $\mathbb{C}$. Then there are (at least) two concepts of fundamental group for $X$: the topological fundamental group $\pi_1^{top}(X)$, i.e. the fundamental group of the underlying topological space $X(\mathbb{C})$ with the analytic topology, and the étale fundamental group $\pi^{et}(X)$, which is a purely algebraic object (it is an inverse limit of finite automorphism groups).

By the so-called Riemann Existence Theorem, the finite covering of $X(\mathbb{C})$, that a priori are only analytic spaces, have actually a scheme structure. This implies that $\pi_1^{et}(X)$ is precisely the profinite completion of $\pi_1^{top}(X)$.

Therefore there is a group homomorphism $$\eta \colon \pi_1^{top}(X) \longrightarrow \pi_1^{et}(X),$$ which is injective precisely when $X$ is residually finite.

For instance, if for some reason we know that $\pi_1^{top}(X)$ is residually finite and we are able to show that $\pi_1^{et}(X)=0,$ then we can conclude that $\pi_1^{top}(X)=0,$ too.

J. P. Serre asked if there exist smooth, complex varieties such that $\pi_1^{top}(X)$ is not residually finite (and hence $\eta$ is not injective). A consequence of a theorem of Malcev is that if $\pi_1^{top}(X)$ has a faithful linear representation, then it is residually finite. Hence if $X$ is a variety answering affirmatively Serre's question, its topological fundamental group must have no faithful linear representation. Such varieties actually exist, and the first examples were constructed by D. Toledo, see

Projective varieties with non-residually finite fundamental group, Publications Mathématiques de l'Institut des Hautes Études Scientifiques Volume 77, Issue 1 (1993), 103-119.

In Toledo's examples, $\ker \eta$ is free group of infinite rank. Later, F. Catanese and J. Kollar, and independently M. Nori, were able to find examples where $\ker \eta$ is a finite cyclic group. Such examples are described in

Classification of irregular varieties, Lecture Notes in Mathematics 1515, Springer 1992.

A typical example arising in algebraic geometry is the following.

Take a scheme $X$ of finite type over $\mathbb{C}$. Then there are (at least) two concepts of fundamental group for $X$: the topological fundamental group $\pi_1^{top}(X)$, i.e. the fundamental group of the underlying topological space $X(\mathbb{C})$ with the analytic topology, and the étale fundamental group $\pi^{et}(X)$, which is a purely algebraic object (it is an inverse limit of finite automorphism groups).

By the so-called Riemann Existence Theorem, the finite covering of $X(\mathbb{C})$, that a priori are only analytic spaces, have actually a scheme structure. This implies that $\pi_1^{et}(X)$ is precisely the profinite completion of $\pi_1^{top}(X)$.

Therefore there is a group homomorphism $$\eta \colon \pi_1^{top}(X) \longrightarrow \pi_1^{et}(X),$$ which is injective precisely when $\pi_1^{top}(X)$ is residually finite.

For instance, if for some reason we know that $\pi_1^{top}(X)$ is residually finite and we are able to show that $\pi_1^{et}(X)=0,$ then we can conclude that $\pi_1^{top}(X)=0,$ too.

J. P. Serre asked if there exist smooth, complex varieties such that $\pi_1^{top}(X)$ is not residually finite (and hence $\eta$ is not injective). A consequence of a theorem of Malcev is that if $\pi_1^{top}(X)$ has a faithful linear representation, then it is residually finite. Hence if $X$ is a variety answering affirmatively Serre's question, its topological fundamental group must have no faithful linear representation. Such varieties actually exist, and the first examples were constructed by D. Toledo, see

Projective varieties with non-residually finite fundamental group, Publications Mathématiques de l'Institut des Hautes Études Scientifiques Volume 77, Issue 1 (1993), 103-119.

In Toledo's examples, $\ker \eta$ is free group of infinite rank. Later, F. Catanese and J. Kollar, and independently M. Nori, were able to find examples where $\ker \eta$ is a finite cyclic group. Such examples are described in

Classification of irregular varieties, Lecture Notes in Mathematics 1515, Springer 1992.

A typical example arising in algebraic geometry is the following.

Take a scheme $X$ of finite type over $\mathbb{C}$. Then there are (at least) two concepts of fundamental group for $X$: the topological fundamental group $\pi_1^{top}(X)$, i.e. the fundamental group of the underlying topological space $X(\mathbb{C})$ with the analytic topology, and the étale fundamental group $\pi^{et}(X)$, which is a purely algebraic object (it is an inverse limit of finite automorphism groups).

By the so-called Riemann Existence Theorem, the finite covering of $X(\mathbb{C})$, that a priori are only analytic spaces, have actually a scheme structure. This implies that $\pi_1^{et}(X)$ is precisely the profinite completion of $\pi_1^{top}(X)$.

Therefore there is a group homomorphism $$\eta \colon \pi_1^{top}(X) \longrightarrow \pi_1^{et}(X),$$ which is injective precisely when $X$ is residually finite.

For instance, if for some reason we know that $\pi_1^{top}(X)$ is residually finite and we are able to show that $\pi_1^{et}(X)=0,$ then we can conclude that $\pi_1^{top}(X)=0,$ too.

J. P. Serre asked if there exist smooth, complex varieties such that $\pi_1^{top}(X)$ is not residually finite (and hence $\eta$ is not injective). A consequence of a theorem of Malcev is that if $\pi_1^{top}(X)$ has a faithful linear representation, then it is residually finite. Hence if $X$ is a variety answering affirmatively Serre's question, its topological fundamental group must have no faithful linear representation. Such varieties actually exist, and the first examples were constructed by D. Toledo, see

Projective varieties with non-residually finite fundamental group, Publications Mathématiques de l'Institut des Hautes Études Scientifiques Volume 77, Issue 1 (1993), 103-119.

A typical example arising in algebraic geometry is the following.

Take a scheme $X$ of finite type over $\mathbb{C}$. Then there are (at least) two concepts of fundamental group for $X$: the topological fundamental group $\pi_1^{top}(X)$, i.e. the fundamental group of the underlying topological space $X(\mathbb{C})$ with the analytic topology, and the étale fundamental group $\pi^{et}(X)$, which is a purely algebraic object (it is an inverse limit of finite automorphism groups).

By the so-called Riemann Existence Theorem, the finite covering of $X(\mathbb{C})$, that a priori are only analytic spaces, have actually a scheme structure. This implies that $\pi_1^{et}(X)$ is precisely the profinite completion of $\pi_1^{top}(X)$.

Therefore there is a group homomorphism $$\eta \colon \pi_1^{top}(X) \longrightarrow \pi_1^{et}(X),$$ which is injective precisely when $X$ is residually finite.

For instance, if for some reason we know that $\pi_1^{top}(X)$ is residually finite and we are able to show that $\pi_1^{et}(X)=0,$ then we can conclude that $\pi_1^{top}(X)=0,$ too.

J. P. Serre asked if there exist smooth, complex varieties such that $\pi_1^{top}(X)$ is not residually finite (and hence $\eta$ is not injective). A consequence of a theorem of Malcev is that if $\pi_1^{top}(X)$ has a faithful linear representation, then it is residually finite. Hence if $X$ is a variety answering affirmatively Serre's question, its topological fundamental group must have no faithful linear representation. Such varieties actually exist, and the first examples were constructed by D. Toledo, see

Projective varieties with non-residually finite fundamental group, Publications Mathématiques de l'Institut des Hautes Études Scientifiques Volume 77, Issue 1 (1993), 103-119.

In Toledo's examples, $\ker \eta$ is free group of infinite rank. Later, F. Catanese and J. Kollar, and independently M. Nori, were able to find examples where $\ker \eta$ is a finite cyclic group. Such examples are described in

Classification of irregular varieties, Lecture Notes in Mathematics 1515, Springer 1992.