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Because Goresky and MacPherson are proving such a general and precise version of the theorem, it is a bit difficult to parse. However, in the main case that $X$ is LCI, the point is that if $Z$ everywhere has codimension $\geq c$, then we can choose the general linear section $H$ of that theorem to have codimension $\text{dim}(X)-c+1$, so that $H$ is disjoint from $Z$. Then the integer $\widehat{n}$ of that theorem simply equals $\text{dim}(X)-c$. Thus, applying the theorem first to $X$ and then to $U=X\setminus Z$ (since $H$ is contained in $U$), the result is that the pushforward map on homotopy groups, $$\pi_i(U)\to \pi_i(X),$$ is an isomorphism for $i< \widehat{n}$, and the map is a surjection if $i$ equals $\widehat{n}$. By Hurewicz, the same thing holds if we replace $\pi_i(-)$ by $H_i(-)$. If you use $\mathbb{Q}$-coefficients and then use the Universal Coefficients Theorem, you get the analogous result in cohomology. You can also get cohomology result with torsion coefficients, e.g., for a complex variety $X$ that is LCI, for a Zariski closed subset $Z$ of $X$ whose codimension is everywhere $\geq 4$, the map of Brauer groups from $X$ to $X\setminus Z$ is an isomorphism. (Does anybody know a good example proving this codimension hypothesis is best possible? Quadratic hypersurfaces have trivial Brauer group, so that does not seem to work.)

Thus, for every complex variety $X$ that is LCI and whose singular locus $Z$ of $X$ everywhere has codimension $\geq c$, the pushforward map, $$H_i(X^{\text{smooth}},\mathbb{Q}) \to H_i(X,\mathbb{Q}),$$ is an isomorphism for $i<\text{dim}(X)-c$, and it is surjective if $i=\text{dim}(X)-c$.

Because Goresky and MacPherson are proving such a general and precise version of the theorem, it is a bit difficult to parse. However, in the main case that $X$ is LCI, the point is that if $Z$ everywhere has codimension $\geq c$, then we can choose the general linear section $H$ of that theorem to have codimension $b=\text{dim}(X)-c+1$, so that $H$ is disjoint from $Z$. Then the integer $\widehat{n}$ of that theorem simply equals $\text{dim}(X)-b=c-1$ (Edit this is not $\text{dim}(X)-c$ as previously written). Thus, applying the theorem first to $X$ and then to $U=X\setminus Z$ (since $H$ is contained in $U$), the result is that the pushforward map on homotopy groups, $$\pi_i(U)\to \pi_i(X),$$ is an isomorphism for $i< \widehat{n}=c-1$, and the map is a surjection if $i$ equals $\widehat{n}=c-1$. By Hurewicz, the same thing holds if we replace $\pi_i(-)$ by $H_i(-)$. If you use $\mathbb{Q}$-coefficients and then use the Universal Coefficients Theorem, you get the analogous result in cohomology. You can also get cohomology result with torsion coefficients, e.g., for a complex variety $X$ that is LCI, for a Zariski closed subset $Z$ of $X$ whose codimension is everywhere $\geq 4$, the map of Brauer groups from $X$ to $X\setminus Z$ is an isomorphism. (Does anybody know a good example proving this codimension hypothesis is best possible? Quadratic hypersurfaces have trivial Brauer group, so that does not seem to work.)

Thus, for every complex variety $X$ that is LCI and whose singular locus $Z$ of $X$ everywhere has codimension $\geq c$, the pushforward map, $$H_i(X^{\text{smooth}},\mathbb{Q}) \to H_i(X,\mathbb{Q}),$$ is an isomorphism for $i<c-1$, and it is surjective if $i=c-1$.

Because Goresky and MacPherson are proving such a general and precise version of the theorem, it is a bit difficult to parse. However, in the main case that $X$ is LCI, the point is that if $Z$ everywhere has codimension $\geq c$, then we can choose the general linear section $H$ of that theorem to have codimension $\text{dim}(X)-c+1$, so that $H$ is disjoint from $Z$. Then the integer $\widehat{n}$ of that theorem simply equals $\text{dim}(X)-c$. Thus, applying the theorem first to $X$ and then to $U=X\setminus Z$ (since $H$ is contained in $U$), the result is that the pushforward map on homotopy groups, $$\pi_i(U)\to \pi_i(X),$$ is an isomorphism for $i< \widehat{n}$, and the map is a surjection if $i$ equals $\widehat{n}$. By Hurewicz, the same thing holds if we replace $\pi_i(-)$ by $H_i(-)$. If you use $\mathbb{Q}$-coefficients and then use the Universal Coefficients Theorem, you get the analogous result in cohomology (you can also get cohomology result with torsion coefficients).

Because Goresky and MacPherson are proving such a general and precise version of the theorem, it is a bit difficult to parse. However, in the main case that $X$ is LCI, the point is that if $Z$ everywhere has codimension $\geq c$, then we can choose the general linear section $H$ of that theorem to have codimension $\text{dim}(X)-c+1$, so that $H$ is disjoint from $Z$. Then the integer $\widehat{n}$ of that theorem simply equals $\text{dim}(X)-c$. Thus, applying the theorem first to $X$ and then to $U=X\setminus Z$ (since $H$ is contained in $U$), the result is that the pushforward map on homotopy groups, $$\pi_i(U)\to \pi_i(X),$$ is an isomorphism for $i< \widehat{n}$, and the map is a surjection if $i$ equals $\widehat{n}$. By Hurewicz, the same thing holds if we replace $\pi_i(-)$ by $H_i(-)$. If you use $\mathbb{Q}$-coefficients and then use the Universal Coefficients Theorem, you get the analogous result in cohomology. You can also get cohomology result with torsion coefficients, e.g., for a complex variety $X$ that is LCI, for a Zariski closed subset $Z$ of $X$ whose codimension is everywhere $\geq 4$, the map of Brauer groups from $X$ to $X\setminus Z$ is an isomorphism. (Does anybody know a good example proving this codimension hypothesis is best possible? Quadratic hypersurfaces have trivial Brauer group, so that does not seem to work.)

J. S. Milne
Lectures on étale cohomology
https://www.jmilne.org/math/CourseNotes/LEC.pdf

For low degree cohomology and its algebraic avatars (étale fundamental groups, Picard groups, Brauer groups, ...), there are other Purity Theorems that usually require much less than smoothness. One typical hypothesis is that $X$ is everywhere locally a complete intersection (LCI). Two great references are SGA 2 and Grothendieck's three exposes, "Le groupe de Brauer" in "Dix exposes sur la cohomologie des schemas". First, regarding purity for connectedness, i.e., $\pi_0$, there is $S2$ extension which is greatly generalized in Hartshorne's Connectedness Theorem. This says that if you remove a Zariski closed subset $Z$ from a pure-dimensional variety $X$ that is LCI, or even just $S2$, this does not change $\pi_0$ provided that $Z$ everywhere has codimension $\geq 2$ (regardless of singularities of $X$ and $Z$ beyond the $S2$ hypothesis).

If $X$ is LCI and if the codimension of $Z$ is everywhere at least $3$, then the pushforward map of étale fundamental groups from $U$ to $X$ is an isomorphism.

The next Purity Theorem is for Picard groups (roughly an $H^2$ result rather than the $H^1$ result coming from fundamental groups). The first results were proved by Auslander-Buchsbaum for $X$ smooth (their famous theorem, later considered also by Serre, about factoriality of regular local rings). The LCI case was conjectured by Samuel and proved by Grothendieck: Théorème XI.3.13 and Corollaire XI.3.14 of loc. cit.

Because Goresky and MacPherson are proving such a general and precise version of the theorem, it is a bit difficult to parse. However, in the main case that $X$ is LCI, the point is that if $Z$ everywhere has codimension $\geq c$, then we can choose the general linear section $H$ of that theorem to have codimension $\text{dim}(X)-c+1$, so that $H$ is disjoint from $Z$. Then the integer $\widehat{n}$ of that theorem simply equals $\text{dim}(X)-c$. Thus, applying the theorem first to $X$ and then to $U=X\setminus Z$ (since $H$ is contained in $U$), the result is that the pushforward map on homotopy groups, $$\pi_i(U)\to \pi_i(X)$$ is an isomorphism for $i< \widehat{n}$, and the map is a surjection if $i$ equals $\widehat{n}$. By Hurewicz, the same thing holds if we replace $\pi_i(-)$ by $H_i(-)$. If you use $\mathbb{Q}$-coefficients and then use the Universal Coefficients Theorem, you get the analogous result in cohomology (you can also get cohomology result with torsion coefficients).

Thus, for every complex variety $X$ that is LCI and whose singular locus $Z$ of $X$ everywhere has codimension $\geq c$, the pushforward maps, $$H_i(X^{\text{smooth}},\mathbb{Q}) \to H_i(X,\mathbb{Q}),$$ are isomorphisms for $i<\text{dim}(X)-c$, and it is surjective if $i=\text{dim}(X)-c$.

J. S. Milne
Lectures on étale cohomology
https://www.jmilne.org/math/CourseNotes/LEC.pdf

For low degree cohomology and its algebraic avatars (étale fundamental groups, Picard groups, Brauer groups, ...), there are other Purity Theorems that usually require much less than smoothness. One typical hypothesis is that $X$ is everywhere locally a complete intersection (LCI). Two great references are SGA 2 and Grothendieck's three exposes, "Le groupe de Brauer" in "Dix exposes sur la cohomologie des schemas". First, regarding purity for connectedness, i.e., $\pi_0$, there is $S2$ extension which is greatly generalized in Hartshorne's Connectedness Theorem. This says that if you remove a Zariski closed subset $Z$ from a pure-dimensional variety $X$ that is LCI, or even just $S2$, this does not change $\pi_0$ provided that $Z$ everywhere has codimension $\geq 2$ (regardless of singularities of $X$ and $Z$ beyond the $S2$ hypothesis). The hypothesis on the codimension is necessary -- just consider a quadric hypersurface of rank $2$ and $Z$ is its codimension $1$ singular locus.

If $X$ is LCI and if the codimension of $Z$ is everywhere at least $3$, then the pushforward map of étale fundamental groups from $U$ to $X$ is an isomorphism. The hypothesis on the codimension is necessary -- just consider a quadric hypersurface of rank $3$ and $Z$ is its codimension $2$ singular locus, cf. Exercise II.6.5 in Hartshorne's Algebraic geometry.

The next Purity Theorem is for Picard groups (roughly an $H^2$ result rather than the $H^1$ result coming from fundamental groups). The first results were proved by Auslander-Buchsbaum for $X$ smooth (their famous theorem, later considered also by Serre, about factoriality of regular local rings). The LCI case was conjectured by Samuel and proved by Grothendieck: Théorème XI.3.13 and Corollaire XI.3.14 of loc. cit. Here the hypothesis is that $Z$ has codimension at least $4$.
The hypothesis on the codimension is necessary -- just consider a quadric hypersurface of rank $4$ and $Z$ is its codimension $3$ singular locus, cf. Exercise II.6.5 in Hartshorne's Algebraic geometry.

Because Goresky and MacPherson are proving such a general and precise version of the theorem, it is a bit difficult to parse. However, in the main case that $X$ is LCI, the point is that if $Z$ everywhere has codimension $\geq c$, then we can choose the general linear section $H$ of that theorem to have codimension $\text{dim}(X)-c+1$, so that $H$ is disjoint from $Z$. Then the integer $\widehat{n}$ of that theorem simply equals $\text{dim}(X)-c$. Thus, applying the theorem first to $X$ and then to $U=X\setminus Z$ (since $H$ is contained in $U$), the result is that the pushforward map on homotopy groups, $$\pi_i(U)\to \pi_i(X),$$ is an isomorphism for $i< \widehat{n}$, and the map is a surjection if $i$ equals $\widehat{n}$. By Hurewicz, the same thing holds if we replace $\pi_i(-)$ by $H_i(-)$. If you use $\mathbb{Q}$-coefficients and then use the Universal Coefficients Theorem, you get the analogous result in cohomology (you can also get cohomology result with torsion coefficients).

Thus, for every complex variety $X$ that is LCI and whose singular locus $Z$ of $X$ everywhere has codimension $\geq c$, the pushforward map, $$H_i(X^{\text{smooth}},\mathbb{Q}) \to H_i(X,\mathbb{Q}),$$ is an isomorphism for $i<\text{dim}(X)-c$, and it is surjective if $i=\text{dim}(X)-c$.