Cohomology of real projective space: Difference between revisions

Latest revision as of 21:07, 30 September 2011

This article describes the value (and the process used to compute it) of some homotopy invariant(s) for a topological space or family of topological spaces. The invariant is cohomology group and the topological space/family is real projective space
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Cohomology groups in piecewise form

Odd-dimensional projective space with coefficients in integers

$H^{p} (P^{n} (R); Z) = {\begin{matrix} Z, & p = 0, n \\ Z / 2 Z, & p e v e n, 0 < p < n \\ 0, & o t h e r w i s e \end{matrix}$

Even-dimensional projective space with coefficients in integers

$H^{p} (P^{n} (R); Z) = {\begin{matrix} Z, & p = 0 \\ Z / 2 Z, & p e v e n, 0 < p \leq n \\ 0, & o t h e r w i s e \end{matrix}$

Odd-dimensional projective space with coefficients in an abelian group

For an abelian group $M$ , the cohomology is given by:

$H^{p} (P^{n} (R); M) = {\begin{matrix} M, & p = 0, n \\ M / 2 M, & p e v e n, 0 < p < n \\ T, & p o d d, 0 < p < n \\ 0, & o t h e r w i s e \end{matrix}$

Here, $T$ denotes the 2-torsion subgroup, i.e., the subgroup comprising elements of order dividing 2.

Even-dimensional projective space with coefficients in an abelian group

For an abelian group $M$ , the cohomology is given by:

$H^{p} (P^{n} (R); M) = {\begin{matrix} M, & p = 0 \\ M / 2 M, & p e v e n, 0 < p \leq n \\ T, & p o d d, 0 < p < n \\ 0, & o t h e r w i s e \end{matrix}$

Here, $T$ denotes the 2-torsion subgroup, i.e., the subgroup comprising elements of order dividing 2.

Coefficients in a module over a 2-divisible ring

If we consider the cohomology with coefficients in a module $M$ over a ring $R$ where 2 is invertible, then we have:

$H^{p} (P^{n} (R); M) : = {\begin{matrix} M, & p = 0 \\ M, & p = n, n o d d \\ 0, & p = n, n e v e n \\ 0, & p \neq 0, n \end{matrix}$

In particular, these results are valid over the field of rational numbers or over any field of characteristic zero, or indeed any characteristic other than 2.

Coefficients in characteristic two

Suppose $M$ is an elementary abelian 2-group, i.e., a group in which the double of every element is zero. Then, $2 M = 0$ (so $M / 2 M ≅ M$ ) and $T = M$ , and we get:

$H^{p} (P^{n} (R); M) = {\begin{matrix} M, & 0 \leq p \leq n \\ 0, & o t h e r w i s e \end{matrix}$

This in particular applies to the case that $M$ is the group $Z / 2 Z$ , i.e., when we are taking coefficients in the field of two elements.

Cohomology groups in tabular form

Coefficients in integers

We illustrate how the cohomology groups work for small values of $n$ . Note that for $p > n$ , all cohomology groups $H^{p}$ are zero, so we omit those cells for visual ease.

$n$	Real projective space $R P^{n}$	Orientable?	$H^{0}$	$H^{1}$	$H^{2}$	$H^{3}$	$H^{4}$	$H^{5}$
1	circle	Yes	$Z$	$Z$
2	real projective plane	No	$Z$	0	$Z / 2 Z$
3	RP^3	Yes	$Z$	0	$Z / 2 Z$	$Z$
4	RP^4	No	$Z$	0	$Z / 2 Z$	0	$Z / 2 Z$
5	RP^5	Yes	$Z$	0	$Z / 2 Z$	0	$Z / 2 Z$	$Z$

Coefficients in an abelian group

We let the abelian group be $M$ . Denote by $T$ the 2-torsion of $M$ and by $2 M$ the submodule comprising doubles of elements.

$n$	Real projective space $R P^{n}$	Orientable?	$H^{0}$	$H^{1}$	$H^{2}$	$H^{3}$	$H^{4}$	$H^{5}$
1	circle	Yes	$M$	$M$
2	real projective plane	No	$M$	$T$	$M / 2 M$
3	RP^3	Yes	$M$	$T$	$M / 2 M$	$M$
4	RP^4	No	$M$	$T$	$M / 2 M$	$T$	$M / 2 M$
5	RP^5	Yes	$M$	$T$	$M / 2 M$	$T$	$M / 2 M$	$M$

Coefficients in a module over a 2-divisible ring

Suppose $M$ has the structure of a module over a unital ring $R$ where 2 is invertible. Then, in particular, we know that $2 M = M$ and $T = 0$ . Thus, both $M / 2 M$ and $T$ are equal to $0$ . We get:

$n$	Real projective space $R P^{n}$	Orientable?	$H^{0}$	$H^{1}$	$H^{2}$	$H^{3}$	$H^{4}$	$H^{5}$
1	circle	Yes	$M$	$M$
2	real projective plane	No	$M$	0	0
3	RP^3	Yes	$M$	0	0	$M$
4	RP^4	No	$M$	0	0	0	0
5	RP^5	Yes	$M$	0	0	0	0	$M$

Coefficients in characteristic two

Suppose $M$ is an elementary abelian 2-group, i.e., a group in which the double of every element is zero. Then, $2 M = 0$ (so $M / 2 M ≅ M$ ) and $T = M$ , and we get:

$n$	Real projective space $R P^{n}$	Orientable?	$H^{0}$	$H^{1}$	$H^{2}$	$H^{3}$	$H^{4}$	$H^{5}$
1	circle	Yes	$M$	$M$
2	real projective plane	No	$M$	$M$	$M$
3	RP^3	Yes	$M$	$M$	$M$	$M$
4	RP^4	No	$M$	$M$	$M$	$M$	$M$
5	RP^5	Yes	$M$	$M$	$M$	$M$	$M$	$M$

Cohomology ring structure

Over the integers for even $n$

The cohomology ring $H^{*} (R P^{n}; Z)$ is the ring $Z [x] / ⟨ 2 x, x^{(n / 2) + 1} ⟩$ , where $x$ is the unique non-identity element in $H^{2} (R P^{n}; Z)$ . $x^{r}$ in turn is the unique non-identity element in $H^{2 r} (R P^{n}; Z)$ for $1 \leq r \leq n / 2$ . The coefficients ring (i.e., the constant terms) is $H^{0}$ .

Note that that is almost the same as the ring $Z / 2 Z [x] / ⟨ x^{(n / 2) + 1} ⟩$ , with the only difference being that for the constant terms, we are allowed to use the ring $Z$ rather than the quotient ring $Z / 2 Z$ .

Over the integers for odd $n$

The cohomology ring $H^{*} (R P^{n}; Z)$ is the ring $Z [x, y] / ⟨ 2 x, x^{(n + 1) / 2}, x y, y^{2} ⟩$ where $x$ is the unique non-identity element in $H^{2} (R P^{n}; Z)$ and $y$ is a generator of $H^{n} (R P^{n}; Z)$ . $x^{r}$ in turn is the unique non-identity element in $H^{2 r} (R P^{n}; Z)$ for $1 \leq r \leq (n - 1) / 2$ . The coefficients ring (i.e., the constant terms) is $H^{0}$ .

Note that that is almost the same as the ring $Z / 2 Z [x, y] / ⟨ x^{(n / 2) + 1}, y^{2}, x y ⟩$ , with the only difference being that for the constant terms, we are allowed to use the ring $Z$ rather than the quotient ring $Z / 2 Z$ .

Reality checks

General assertion	Verification in this case	See also ...
For any compact connected $n$ -dimensional manifold, the top cohomology group $H^{n}$ is $Z$ if the space is orientable and is (?) (finite group?) otherwise.	$n$ odd: In this case, the space is obtained by taking the quotient of the orientable manifold $S^{n}$ by the antipodal action, which is orientation-preserving (one way of seeing it is that is given by a scalar matrix of $- 1$ s in dimension $n + 1$ , so has determinant 1). The quotient is thus also orientable. Indeed, for $n$ odd, the top cohomology is $Z$ . $n$ even: In this case, the space is obtained by taking the quotient of the orientable manifold $S^{n}$ by the antipodal action, which is orientation-reversing (one way of seeing it is that is given by a scalar matrix of $- 1$ s in dimension $n + 1$ , so has determinant -1). The quotient is thus non-orientable. Indeed, for $n$ even, the top cohomology is $Z / 2 Z$ .	?
For a compact connected orientable manifold of dimension $n$ , the Poincare duality theorem says that the homology group of dimension $p$ is isomorphic to the cohomology group of dimension $n - p$ .	Case $n$ odd: As noted above, the manifold is orientable. The top and bottom homology and cohomology groups are isomorphic to $Z$ . The even-dimensional cohomology groups and odd-dimensional homology groups are both isomorphic to $Z / 2 Z$ . The odd-dimensional cohomology groups and even-dimensional homology groups are both zero groups.	homology of real projective space

@@ Line 1: / Line 1: @@
 {{homotopy invariant computation|
-invariant = homology group|
+invariant = cohomology group|
 space = real projective space}}
-==Statement==
+==Cohomology groups in piecewise form==
 ===Odd-dimensional projective space with coefficients in integers===
-<math>H_p(\mathbb{P}^n(\R)) = \left\lbrace \begin{array}{rl} \Z, & \qquad p=0,n\\ \Z/2\Z, &\qquad p \ \operatorname{even}, 0 < p < n\\ 0, & \qquad \operatorname{otherwise}\end{array}\right.</math>
+<math>H^p(\mathbb{P}^n(\R); \mathbb{Z}) = \left\lbrace \begin{array}{rl} \Z, & \qquad p=0,n\\ \Z/2\Z, &\qquad p \ \operatorname{even}, 0 < p < n\\ 0, & \qquad \operatorname{otherwise}\end{array}\right.</math>
 ===Even-dimensional projective space with coefficients in integers===
-<math>H_p(\mathbb{P}^n(\R)) = \left\lbrace \begin{array}{rl} \Z, & \qquad p=0\\ \Z/2\Z, & \qquad p \ \operatorname{even}, 0 < p \le n\\ 0, & \qquad \operatorname{otherwise}\end{array}\right.</math>
+<math>H^p(\mathbb{P}^n(\R); \mathbb{Z}) = \left\lbrace \begin{array}{rl} \Z, & \qquad p=0\\ \Z/2\Z, & \qquad p \ \operatorname{even}, 0 < p \le n\\ 0, & \qquad \operatorname{otherwise}\end{array}\right.</math>
+===Odd-dimensional projective space with coefficients in an abelian group===
+For an abelian group <math>M</math>, the cohomology is given by:
+<math>H^p(\mathbb{P}^n(\R); M) = \left\lbrace \begin{array}{rl} M, & \qquad p=0,n\\ M/2M, &\qquad p \ \operatorname{even}, 0 < p < n\\ T, & \qquad p \ \operatorname{odd}, 0 < p < n \\ 0, & \qquad \operatorname{otherwise}\end{array}\right.</math>
+Here, <math>T</math> denotes the 2-torsion subgroup, i.e., the subgroup comprising elements of order dividing 2.
+===Even-dimensional projective space with coefficients in an abelian group===
+For an abelian group <math>M</math>, the cohomology is given by:
+<math>H^p(\mathbb{P}^n(\R); M) = \left\lbrace \begin{array}{rl} M, & \qquad p=0\\ M/2M, & \qquad p \ \operatorname{even}, 0 < p \le n\\ T, & \qquad p \ \operatorname{odd}, 0 < p < n \\ 0, & \qquad \operatorname{otherwise}\end{array}\right.</math>
+Here, <math>T</math> denotes the 2-torsion subgroup, i.e., the subgroup comprising elements of order dividing 2.
+===Coefficients in a module over a 2-divisible ring===
+If we consider the cohomology with coefficients in a module <math>M</math> over a ring <math>R</math> where 2 is invertible, then we have:
+<math>H^p(\mathbb{P}^n(\R);M) := \left\lbrace\begin{array}{rl} M, & p = 0 \\ M, & p = n, n \ \operatorname{odd}\\ 0, & p = n, n \ \operatorname{even}\\ 0, & p \ne 0,n \\\end{array}\right.</math>
+In particular, these results are valid over the field of rational numbers or over any field of characteristic zero, or indeed any characteristic other than 2.
+===Coefficients in characteristic two===
+Suppose <math>M</math> is an elementary abelian 2-group, i.e., a group in which the double of every element is zero. Then, <math>2M = 0</math> (so <math>M/2M \cong M</math>) and <math>T = M</math>,  and we get:
+<math>H^p(\mathbb{P}^n(\R);M) = \left\lbrace\begin{array}{rl}M, & \qquad 0 \le p \le n \\ 0, & \qquad \operatorname{otherwise}\\\end{array}\right.</math>
+This in particular applies to the case that <math>M</math> is the group <math>\mathbb{Z}/2\mathbb{Z}</math>, i.e., when we are taking coefficients in the field of two elements.
+==Cohomology groups in tabular form==
+===Coefficients in integers===
+We illustrate how the cohomology groups work for small values of <math>n</math>. Note that for <math>p > n</math>, all cohomology groups <math>H^p</math> are zero, so we omit those cells for visual ease.
+{| class="sortable" border="1"
+! <math>n</math> !! Real projective space <math>\R\mathbb{P}^n</math> !! Orientable? !! <math>H^0</math> !! <math>H^1</math> !! <math>H^2</math> !! <math>H^3</math> !! <math>H^4</math> !! <math>H^5</math>
+|-
+| 1 || [[circle]] || Yes || <math>\mathbb{Z}</math> || <math>\mathbb{Z}</math>
+|-
+| 2 || [[real projective plane]] || No || <math>\mathbb{Z}</math> || 0 || <math>\mathbb{Z}/2\mathbb{Z}</math>
+|-
+| 3 || [[RP^3]] || Yes || <math>\mathbb{Z}</math> || 0 || <math>\mathbb{Z}/2\mathbb{Z}</math>  || <math>\mathbb{Z}</math>
+|-
+| 4 || [[RP^4]] || No || <math>\mathbb{Z}</math> || 0 || <math>\mathbb{Z}/2\mathbb{Z}</math> || 0 || <math>\mathbb{Z}/2\mathbb{Z}</math>
+|-
+| 5 || [[RP^5]] || Yes || <math>\mathbb{Z}</math> || 0 || <math>\mathbb{Z}/2\mathbb{Z}</math> || 0 || <math>\mathbb{Z}/2\mathbb{Z}</math> || <math>\mathbb{Z}</math>
+|}
+===Coefficients in an abelian group===
+We let the abelian group be <math>M</math>. Denote by <math>T</math> the 2-torsion of <math>M</math> and by <math>2M</math> the submodule comprising doubles of elements.
+{| class="sortable" border="1"
+! <math>n</math> !! Real projective space <math>\R\mathbb{P}^n</math> !! Orientable? !! <math>H^0</math> !! <math>H^1</math> !! <math>H^2</math> !! <math>H^3</math> !! <math>H^4</math> !! <math>H^5</math>
+|-
+| 1 || [[circle]] || Yes || <math>M</math> || <math>M</math>
+|-
+| 2 || [[real projective plane]] || No || <math>M</math> || <math>T</math> || <math>M/2M</math>
+|-
+| 3 || [[RP^3]] || Yes || <math>M</math> || <math>T</math> || <math>M/2M</math> || <math>M</math>
+|-
+| 4 || [[RP^4]] || No || <math>M</math> || <math>T</math> || <math>M/2M</math> || <math>T</math> || <math>M/2M</math>
+|-
+| 5 || [[RP^5]] || Yes || <math>M</math> || <math>T</math> || <math>M/2M</math> || <math>T</math> || <math>M/2M</math> || <math>M</math>
+|}
+===Coefficients in a module over a 2-divisible ring===
+Suppose <math>M</math> has the structure of a module over a unital ring <math>R</math> where 2 is invertible. Then, in particular, we know that <math>2M = M</math> and <math>T = 0</math>. Thus, both <math>M/2M</math> and <math>T</math> are equal to <math>0</math>. We get:
+{| class="sortable" border="1"
+! <math>n</math> !! Real projective space <math>\R\mathbb{P}^n</math> !! Orientable? !! <math>H^0</math> !! <math>H^1</math> !! <math>H^2</math> !! <math>H^3</math> !! <math>H^4</math> !! <math>H^5</math>
+|-
+| 1 || [[circle]] || Yes || <math>M</math> || <math>M</math>
+|-
+| 2 || [[real projective plane]] || No || <math>M</math> || 0 || 0
+|-
+| 3 || [[RP^3]] || Yes || <math>M</math> || 0 || 0 || <math>M</math>
+|-
+| 4 || [[RP^4]] || No || <math>M</math> || 0 || 0 || 0 || 0
+|-
+| 5 || [[RP^5]] || Yes || <math>M</math> || 0 || 0 || 0 || 0 || <math>M</math>
+|}
+===Coefficients in characteristic two===
+Suppose <math>M</math> is an elementary abelian 2-group, i.e., a group in which the double of every element is zero. Then, <math>2M = 0</math> (so <math>M/2M \cong M</math>) and <math>T = M</math>,  and we get:
+{| class="sortable" border="1"
+! <math>n</math> !! Real projective space <math>\R\mathbb{P}^n</math> !! Orientable? !! <math>H^0</math> !! <math>H^1</math> !! <math>H^2</math> !! <math>H^3</math> !! <math>H^4</math> !! <math>H^5</math>
+|-
+| 1 || [[circle]] || Yes || <math>M</math> || <math>M</math>
+|-
+| 2 || [[real projective plane]] || No || <math>M</math> || <math>M</math> || <math>M</math>
+|-
+| 3 || [[RP^3]] || Yes || <math>M</math> || <math>M</math> || <math>M</math> || <math>M</math>
+|-
+| 4 || [[RP^4]] || No || <math>M</math> || <math>M</math> || <math>M</math> || <math>M</math> || <math>M</math>
+|-
+| 5 || [[RP^5]] || Yes || <math>M</math> || <math>M</math> || <math>M</math> || <math>M</math>|| <math>M</math>|| <math>M</math>
+|}
+==Cohomology ring structure==
+===Over the integers for even <math>n</math>===
+The cohomology ring <math>H^*(\mathbb{R}\mathbb{P}^n;\mathbb{Z})</math> is the ring <math>\mathbb{Z}[x]/\langle 2x, x^{(n/2)+1} \rangle</math>, where <math>x</math> is the unique non-identity element in <math>H^2(\mathbb{R}\mathbb{P}^n;\mathbb{Z})</math>. <math>x^r</math> in turn is the unique non-identity element in <math>H^{2r}(\mathbb{R}\mathbb{P}^n;\mathbb{Z})</math> for <math>1 \le r \le n/2</math>. The coefficients ring (i.e., the constant terms) is <math>H^0</math>.
+Note that that is ''almost'' the same as the ring <math>\mathbb{Z}/2\mathbb{Z}[x]/\langle x^{(n/2)+1} \rangle</math>, with the only difference being that for the constant terms, we are allowed to use the ring <math>\mathbb{Z}</math> rather than the quotient ring <math>\mathbb{Z}/2\mathbb{Z}</math>.
+===Over the integers for odd <math>n</math>===
+The cohomology ring <math>H^*(\mathbb{R}\mathbb{P}^n;\mathbb{Z})</math> is the ring <math>\mathbb{Z}[x,y]/\langle 2x, x^{(n+1)/2}, xy, y^2\rangle</math> where <math>x</math> is the unique non-identity element in <math>H^2(\mathbb{R}\mathbb{P}^n;\mathbb{Z})</math> and <math>y</math> is a generator of <math>H^n(\mathbb{R}\mathbb{P}^n;\mathbb{Z})</math>. <math>x^r</math> in turn is the unique non-identity element in <math>H^{2r}(\mathbb{R}\mathbb{P}^n;\mathbb{Z})</math> for <math>1 \le r \le (n-1)/2</math>. The coefficients ring (i.e., the constant terms) is <math>H^0</math>.
+Note that that is ''almost'' the same as the ring <math>\mathbb{Z}/2\mathbb{Z}[x,y]/\langle x^{(n/2)+1},y^2,xy \rangle</math>, with the only difference being that for the constant terms, we are allowed to use the ring <math>\mathbb{Z}</math> rather than the quotient ring <math>\mathbb{Z}/2\mathbb{Z}</math>.
+==Reality checks==
+{| class="sortable" border="1"
+! General assertion !! Verification in this case !! See also ...
+|-
+| For any compact connected <math>n</math>-dimensional manifold, the top cohomology group <math>H^n</math> is <math>\mathbb{Z}</math> if the space is orientable and is (?) (finite group?) otherwise. || <math>n</math> odd: In this case, the space is obtained by taking the quotient of the orientable manifold <math>S^n</math> by the antipodal action, which is orientation-preserving (one way of seeing it is that is given by a scalar matrix of <math>-1</math>s in dimension <math>n + 1</math>, so has determinant 1). The quotient is thus also orientable. Indeed, for <math>n</math> odd, the top cohomology is <math>\mathbb{Z}</math>.<br><math>n</math> even: In this case, the space is obtained by taking the quotient of the orientable manifold <math>S^n</math> by the antipodal action, which is orientation-reversing (one way of seeing it is that is given by a scalar matrix of <math>-1</math>s in dimension <math>n + 1</math>, so has determinant -1). The quotient is thus non-orientable. Indeed, for <math>n</math> even, the top cohomology is <math>\mathbb{Z}/2\mathbb{Z}</math>. || ?
+|-
+| For a compact connected orientable manifold of dimension <math>n</math>, the [[Poincare duality theorem]] says that the homology group of dimension <math>p</math> is isomorphic to the cohomology group of dimension <math>n - p</math>. || Case <math>n</math> odd: As noted above, the manifold is orientable. The top and bottom homology and cohomology groups are isomorphic to <math>\mathbb{Z}</math>. The even-dimensional cohomology groups and odd-dimensional homology groups are both isomorphic to <math>\mathbb{Z}/2\mathbb{Z}</math>. The odd-dimensional cohomology groups and even-dimensional homology groups are both zero groups. || [[homology of real projective space]]
+|}
+==Facts used==
+# [[uses::Homology of real projective space]]
+# [[uses::Dual universal coefficients theorem]]
+# [[uses::CW structure of real projective space]]
+==Proof using homology groups==
+===Case of odd dimension===
+{{fillin}}
+===Case of even dimension===
+{{fillin}}
+==Proof using cochain complex constructed from CW structure==
+{{fillin}}

Latest revision as of 21:07, 30 September 2011

Cohomology groups in piecewise form

Odd-dimensional projective space with coefficients in integers

Even-dimensional projective space with coefficients in integers

Odd-dimensional projective space with coefficients in an abelian group

Even-dimensional projective space with coefficients in an abelian group

Coefficients in a module over a 2-divisible ring

Coefficients in characteristic two

Cohomology groups in tabular form

Coefficients in integers

Coefficients in an abelian group

Coefficients in a module over a 2-divisible ring

Coefficients in characteristic two

Cohomology ring structure

Over the integers for even n

Over the integers for odd n

Reality checks

Facts used

Proof using homology groups

Case of odd dimension

Case of even dimension

Proof using cochain complex constructed from CW structure

Over the integers for even $n$

Over the integers for odd $n$