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Let $M$ be a compact oriented $n$-manifold. Denote $\Omega^k := {\bigwedge}^k T^*M$ the vector bundle of differential $k$-forms, and let $\Omega^{\text{odd}} := \bigoplus_{\text{$k$ odd}} \Omega^k$ and $\Omega^{\text{even}} := \bigoplus_{\text{$k$ even}} \Omega^k$ be the bundles of odd and even differential forms, respectively.

When are $\Omega^{\text{odd}}$ and $\Omega^{\text{even}}$ isomorphic as vector bundles over $M$?

A few observations:

• If $n$ is odd they are isomorphic, $\Omega^{\text{odd}} \simeq \Omega^{\text{even}}$, as can be seen e.g. by introducing a Riemannian metric $g$ on $M$, and observing that the Hodge star $\star: \Omega^k \to \Omega^{n - k}$ is an isomorphism for eack $k$ and hence also between $\Omega^{\text{odd}}$ and $\Omega^{\text{even}}$.

• More generally, if $\chi(M) = 0$, then $\Omega^{\text{odd}} \simeq \Omega^{\text{even}}$. Proof: there is a non-vanishing section $s$ of $\Omega^1 \simeq TM$. Write $\Omega^1 = \mathbb{R}s \oplus V$ for some vector bundle $V$. Then $$\Omega^{\text{odd}} = s \wedge \Omega^{\text{even}}(V) \oplus \Omega^{\text{odd}}(V) \simeq \Omega(V) \simeq s \wedge \Omega^{\text{odd}}(V) \oplus \Omega^{\mathrm{text}}(V) = \Omega^{\mathrm{text}},$$ where $\Omega(V)$ is the full exterior bundle of $V$, and $\Omega^{\text{odd/even}}(V)$ are the odd/even parts of $\Omega(V)$.

• However, if $n = 2$ they are not isomorphic unless $M$ is the torus: the Euler class of $\Omega^{\text{odd}} = \Omega^1$ is then non-zero, while $\Omega^{\text{even}} = \Omega^0 \oplus \Omega^2 \simeq \mathbb{R}^2$ has trivial Euler class.

• If $n = 2d$, $\Omega^{\text{even}}$ and $\Omega^{\text{odd}}$ can be seen to have identical Chern classes $c_i$, except possibly the top one $c_d$. (I define the Chern class of a real vector bundle to be the Chern class of its complexification.) Namely, if $SM \subset TM$ denotes the unit sphere bundle, and $\pi: SM \to M$ is the footpoint projection, then the pullbacks $\pi^*\Omega^{\text{odd}} \simeq \pi^*\Omega^{\text{even}}$ are isomorphic. (Proof: since there is a non-vanishing tautological section $\tau(x, v) := g_x(v, \bullet)$ of $\Omega^1$, we may repeat the argument in the second point.) By Gysin sequence, $\pi^*$ is an isomorphism on $H^\bullet(M)$ for $\bullet \leq n - 1$, so $c_i(\Omega^{\text{odd}}) = c_i(\Omega^{\text{even}})$ for $i \leq d - 1$, and $c_d(\Omega^{\text{odd}}) - c_d(\Omega^{\text{even}})$ is a multiple of $\chi(M)$ in $H^{2d}(M, \mathbb{Z})$.

• If $n = 4$, my computations suggest that $\Omega^{\text{even}}$ and $\Omega^{\text{odd}}$ have equal all Chern, and Stiefel–Whitney characteristic classes. One needs to use that $\Omega^2 = \Omega^1 \wedge \Omega^1$ and the formula for $c_2$ of a wedge product. In particular, $c_2(\Omega^{\text{odd}}) - c_2(\Omega^{\text{even}}) = 0$ is the zero multiple of the Euler characteristic.

• My guess after all of this is that $\Omega^{\text{even}} \not \simeq \Omega^{\text{odd}}$ if $n$ is even and $\chi(M) \neq 0$, but I was not able to prove this.

Let $M$ be a compact oriented $n$-manifold. Denote $\Omega^k := {\bigwedge}^k T^*M$ the vector bundle of differential $k$-forms, and let $\Omega^{\text{odd}} := \bigoplus_{\text{$k$ odd}} \Omega^k$ and $\Omega^{\text{even}} := \bigoplus_{\text{$k$ even}} \Omega^k$ be the bundles of odd and even differential forms, respectively.

When are $\Omega^{\text{odd}}$ and $\Omega^{\text{even}}$ isomorphic as vector bundles over $M$?

A few observations:

• If $n$ is odd they are isomorphic, $\Omega^{\text{odd}} \simeq \Omega^{\text{even}}$, as can be seen e.g. by introducing a Riemannian metric $g$ on $M$, and observing that the Hodge star $\star: \Omega^k \to \Omega^{n - k}$ is an isomorphism for eack $k$ and hence also between $\Omega^{\text{odd}}$ and $\Omega^{\text{even}}$.

• More generally, if $\chi(M) = 0$, then $\Omega^{\text{odd}} \simeq \Omega^{\text{even}}$. Proof: there is a non-vanishing section $s$ of $\Omega^1 \simeq TM$. Write $\Omega^1 = \mathbb{R}s \oplus V$ for some vector bundle $V$. Then $$\Omega^{\text{odd}} = s \wedge \Omega^{\text{even}}(V) \oplus \Omega^{\text{odd}}(V) \simeq \Omega(V) \simeq s \wedge \Omega^{\text{odd}}(V) \oplus \Omega^{\text{even}}(V) = \Omega^{\text{even}},$$ where $\Omega(V)$ is the full exterior bundle of $V$, and $\Omega^{\text{odd/even}}(V)$ are the odd/even parts of $\Omega(V)$.

• However, if $n = 2$ they are not isomorphic unless $M$ is the torus: the Euler class of $\Omega^{\text{odd}} = \Omega^1$ is then non-zero, while $\Omega^{\text{even}} = \Omega^0 \oplus \Omega^2 \simeq \mathbb{R}^2$ has trivial Euler class.

• If $n = 2d$, $\Omega^{\text{even}}$ and $\Omega^{\text{odd}}$ can be seen to have identical Chern classes $c_i$, except possibly the top one $c_d$. (I define the Chern class of a real vector bundle to be the Chern class of its complexification.) Namely, if $SM \subset TM$ denotes the unit sphere bundle, and $\pi: SM \to M$ is the footpoint projection, then the pullbacks $\pi^*\Omega^{\text{odd}} \simeq \pi^*\Omega^{\text{even}}$ are isomorphic. (Proof: since there is a non-vanishing tautological section $\tau(x, v) := g_x(v, \bullet)$ of $\Omega^1$, we may repeat the argument in the second point.) By Gysin sequence, $\pi^*$ is an isomorphism on $H^\bullet(M)$ for $\bullet \leq n - 1$, so $c_i(\Omega^{\text{odd}}) = c_i(\Omega^{\text{even}})$ for $i \leq d - 1$, and $c_d(\Omega^{\text{odd}}) - c_d(\Omega^{\text{even}})$ is a multiple of $\chi(M)$ in $H^{2d}(M, \mathbb{Z})$.

• If $n = 4$, my computations suggest that $\Omega^{\text{even}}$ and $\Omega^{\text{odd}}$ have equal all Chern, and Stiefel–Whitney characteristic classes. One needs to use that $\Omega^2 = \Omega^1 \wedge \Omega^1$ and the formula for $c_2$ of a wedge product. In particular, $c_2(\Omega^{\text{odd}}) - c_2(\Omega^{\text{even}}) = 0$ is the zero multiple of the Euler characteristic.

• My guess after all of this is that $\Omega^{\text{even}} \not \simeq \Omega^{\text{odd}}$ if $n$ is even and $\chi(M) \neq 0$, but I was not able to prove this.

Let $M$ be a compact oriented $n$-manifold. Denote $\Omega^k := \Lambda^k T^*M$ the vector bundle of differential $k$-forms, and let $\Omega^{\mathrm{odd}} := \oplus_{k\,\,\mathrm{odd}} \Omega^k$ and $\Omega^{\mathrm{even}} := \oplus_{k\,\,\mathrm{even}} \Omega^k$ be the bundles of odd and even differential forms, respectively.

When are $\Omega^{\mathrm{odd}}$ and $\Omega^{\mathrm{even}}$ isomorphic as vector bundles over $M$?

Few observations:

• If $n$ is odd they are isomorphic, $\Omega^{\mathrm{odd}} \simeq \Omega^{\mathrm{even}}$, as can be seen e.g. by introducing a Riemannian metric $g$ on $M$, and observing that the Hodge star $\star: \Omega^k \to \Omega^{n - k}$ is an isomorphism for eack $k$ and hence also between $\Omega^{\mathrm{odd}}$ and $\Omega^{\mathrm{even}}$.

• More generally, if $\chi(M) = 0$, then $\Omega^{\mathrm{odd}} \simeq \Omega^{\mathrm{even}}$. Proof: there is a non-vanishing section $s$ of $\Omega^1 \simeq TM$. Write $\Omega^1 = \mathbb{R}s \oplus V$ for some vector bundle $V$. Then $$\Omega^{\mathrm{odd}} = s \wedge \Omega^{\mathrm{even}}(V) \oplus \Omega^{\mathrm{odd}}(V) \simeq \Omega(V) \simeq s \wedge \Omega^{\mathrm{odd}}(V) \oplus \Omega^{\mathrm{even}}(V) = \Omega^{\mathrm{even}},$$ where $\Omega(V)$ is the full exterior bundle of $V$, and $\Omega^{\mathrm{odd}/\mathrm{even}}(V)$ are the odd/even parts of $\Omega(V)$.

• However, if $n = 2$ they are not isomorphic unless $M$ is the torus: the Euler class of $\Omega^{\mathrm{odd}} = \Omega^1$ is then non-zero, while $\Omega^{\mathrm{even}} = \Omega^0 \oplus \Omega^2 \simeq \mathbb{R}^2$ has trivial Euler class.

• If $n = 2d$, $\Omega^{\mathrm{even}}$ and $\Omega^{\mathrm{odd}}$ can be seen to have identical Chern classes $c_i$, except possibly the top one $c_d$. (I define the Chern class of a real vector bundle to be the Chern class of its complexification.) Namely, if $SM \subset TM$ denotes the unit sphere bundle, and $\pi: SM \to M$ is the footpoint projection, then the pullbacks $\pi^*\Omega^{\mathrm{odd}} \simeq \pi^*\Omega^{\mathrm{even}}$ are isomorphic. (Proof: since there is a non-vanishing tautological section $\tau(x, v) := g_x(v, \bullet)$ of $\Omega^1$, we may repeat the argument in the second point.) By Gysin sequence, $\pi^*$ is an isomorphism on $H^\bullet(M)$ for $\bullet \leq n - 1$, so $c_i(\Omega^{\mathrm{odd}}) = c_i(\Omega^{\mathrm{even}})$ for $i \leq d - 1$, and $c_d(\Omega^{\mathrm{odd}}) - c_d(\Omega^{\mathrm{even}})$ is a multiple of $\chi(M)$ in $H^{2d}(M, \mathbb{Z})$.

• If $n = 4$, my computations suggest that $\Omega^{\mathrm{even}}$ and $\Omega^{\mathrm{odd}}$ have equal all Chern, and Stiefel-Whitney characteristic classes. One needs to use that $\Omega^2 = \Omega^1 \wedge \Omega^1$ and the formula for $c_2$ of a wedge product. In particular, $c_2(\Omega^{\mathrm{odd}}) - c_2(\Omega^{\mathrm{even}}) = 0$ is the $\it{zero\,\, multiple}$ of the Euler characteristic.

• My guess after all of this is that $\Omega^{\mathrm{even}} \not \simeq \Omega^{\mathrm{odd}}$ if $n$ is even and $\chi(M) \neq 0$, but I was not able to prove this.

Let $M$ be a compact oriented $n$-manifold. Denote $\Omega^k := {\bigwedge}^k T^*M$ the vector bundle of differential $k$-forms, and let $\Omega^{\text{odd}} := \bigoplus_{\text{$k$ odd}} \Omega^k$ and $\Omega^{\text{even}} := \bigoplus_{\text{$k$ even}} \Omega^k$ be the bundles of odd and even differential forms, respectively.

When are $\Omega^{\text{odd}}$ and $\Omega^{\text{even}}$ isomorphic as vector bundles over $M$?

A few observations:

• If $n$ is odd they are isomorphic, $\Omega^{\text{odd}} \simeq \Omega^{\text{even}}$, as can be seen e.g. by introducing a Riemannian metric $g$ on $M$, and observing that the Hodge star $\star: \Omega^k \to \Omega^{n - k}$ is an isomorphism for eack $k$ and hence also between $\Omega^{\text{odd}}$ and $\Omega^{\text{even}}$.

• More generally, if $\chi(M) = 0$, then $\Omega^{\text{odd}} \simeq \Omega^{\text{even}}$. Proof: there is a non-vanishing section $s$ of $\Omega^1 \simeq TM$. Write $\Omega^1 = \mathbb{R}s \oplus V$ for some vector bundle $V$. Then $$\Omega^{\text{odd}} = s \wedge \Omega^{\text{even}}(V) \oplus \Omega^{\text{odd}}(V) \simeq \Omega(V) \simeq s \wedge \Omega^{\text{odd}}(V) \oplus \Omega^{\mathrm{text}}(V) = \Omega^{\mathrm{text}},$$ where $\Omega(V)$ is the full exterior bundle of $V$, and $\Omega^{\text{odd/even}}(V)$ are the odd/even parts of $\Omega(V)$.

• However, if $n = 2$ they are not isomorphic unless $M$ is the torus: the Euler class of $\Omega^{\text{odd}} = \Omega^1$ is then non-zero, while $\Omega^{\text{even}} = \Omega^0 \oplus \Omega^2 \simeq \mathbb{R}^2$ has trivial Euler class.

• If $n = 2d$, $\Omega^{\text{even}}$ and $\Omega^{\text{odd}}$ can be seen to have identical Chern classes $c_i$, except possibly the top one $c_d$. (I define the Chern class of a real vector bundle to be the Chern class of its complexification.) Namely, if $SM \subset TM$ denotes the unit sphere bundle, and $\pi: SM \to M$ is the footpoint projection, then the pullbacks $\pi^*\Omega^{\text{odd}} \simeq \pi^*\Omega^{\text{even}}$ are isomorphic. (Proof: since there is a non-vanishing tautological section $\tau(x, v) := g_x(v, \bullet)$ of $\Omega^1$, we may repeat the argument in the second point.) By Gysin sequence, $\pi^*$ is an isomorphism on $H^\bullet(M)$ for $\bullet \leq n - 1$, so $c_i(\Omega^{\text{odd}}) = c_i(\Omega^{\text{even}})$ for $i \leq d - 1$, and $c_d(\Omega^{\text{odd}}) - c_d(\Omega^{\text{even}})$ is a multiple of $\chi(M)$ in $H^{2d}(M, \mathbb{Z})$.

• If $n = 4$, my computations suggest that $\Omega^{\text{even}}$ and $\Omega^{\text{odd}}$ have equal all Chern, and Stiefel–Whitney characteristic classes. One needs to use that $\Omega^2 = \Omega^1 \wedge \Omega^1$ and the formula for $c_2$ of a wedge product. In particular, $c_2(\Omega^{\text{odd}}) - c_2(\Omega^{\text{even}}) = 0$ is the zero multiple of the Euler characteristic.

• My guess after all of this is that $\Omega^{\text{even}} \not \simeq \Omega^{\text{odd}}$ if $n$ is even and $\chi(M) \neq 0$, but I was not able to prove this.

Let $M$ be a compact oriented $n$-manifold. Denote $\Omega^k := \Lambda^k T^*M$ the vector bundle of differential $k$-forms, and let $\Omega^{\mathrm{odd}} := \oplus_{k\,\,\mathrm{odd}} \Omega^k$ and $\Omega^{\mathrm{even}} := \oplus_{k\,\,\mathrm{even}} \Omega^k$ be the bundles of odd and even differential forms, respectively. When are $\Omega^{\mathrm{odd}}$ and $\Omega^{\mathrm{even}}$ isomorphic as vector bundles over $M$?

Few observations:

• If $n$ is odd $\Omega^{\mathrm{odd}} \simeq \Omega^{\mathrm{even}}$, as can be seen e.g. by introducing a Riemannian metric $g$ on $M$, and observing that the Hodge star $\star: \Omega^k \to \Omega^{n - k}$ is an isomorphism for eack $k$ and hence also between $\Omega^{\mathrm{odd}}$ and $\Omega^{\mathrm{even}}$.

• More generally, if $\chi(M) = 0$, then $\Omega^{\mathrm{odd}} \simeq \Omega^{\mathrm{even}}$. Proof: there is a non-vanishing section $s$ of $\Omega^1 \simeq TM$. Write $\Omega^1 = \mathbb{R}s \oplus V$ for some vector bundle $V$. Then $$\Omega^{\mathrm{odd}} = s \wedge \Omega^{\mathrm{even}}(V) \oplus \Omega^{\mathrm{odd}}(V) \simeq \Omega(V) \simeq s \wedge \Omega^{\mathrm{odd}}(V) \oplus \Omega^{\mathrm{even}}(V) = \Omega^{\mathrm{even}},$$ where $\Omega(V)$ is the full exterior bundle of $V$, and $\Omega^{\mathrm{odd}/\mathrm{even}}(V)$ are the odd/even parts of $\Omega(V)$.

• However, if $n = 2$ they are not isomorphic if $M$ is not the torus: the Euler class of $\Omega^{\mathrm{odd}} = \Omega^1$ is then non-zero, while $\Omega^{\mathrm{even}} = \Omega^0 \oplus \Omega^2 \simeq \mathbb{R}^2$ has trivial Euler class.

• If $n = 2d$, $\Omega^{\mathrm{even}}$ and $\Omega^{\mathrm{odd}}$ can be seen to have identical Chern classes $c_i$, except possibly the top one $c_d$. (I define the Chern class of a real vector bundle to be the Chern class of its complexification.) Namely, if $SM \subset TM$ denotes the unit sphere bundle, and $\pi: SM \to M$ is the footpoint projection, then the pullbacks $\pi^*\Omega^{\mathrm{odd}} \simeq \pi^*\Omega^{\mathrm{even}}$ are isomorphic. (Proof: since there is a non-vanishing tautological section $\tau(x, v) := g_x(v, \bullet)$ of $\Omega^1$, we may repeat the argument in the second point.) By Gysin sequence, $\pi^*$ is an isomorphism on $H^\bullet(M)$ for $\bullet \leq n - 1$, so $c_i(\Omega^{\mathrm{odd}}) = c_i(\Omega^{\mathrm{even}})$ for $i \leq d - 1$, and $c_d(\Omega^{\mathrm{odd}}) - c_d(\Omega^{\mathrm{even}})$ is a multiple of $\chi(M)$ in $H^{2d}(M, \mathbb{Z})$.

• If $n = 4$, my computations suggest that $\Omega^{\mathrm{even}}$ and $\Omega^{\mathrm{odd}}$ have equal all Chern, and Stiefel-Whitney characteristic classes. One needs to use that $\Omega^2 = \Omega^1 \wedge \Omega^1$ and the formula for $c_2$ of a wedge product. In particular, $c_2(\Omega^{\mathrm{odd}}) - c_2(\Omega^{\mathrm{even}}) = 0$ is the $\it{zero\,\, multiple}$ of the Euler characteristic.

• My guess after all of this is that $\Omega^{\mathrm{even}} \not \simeq \Omega^{\mathrm{odd}}$ if $n$ is even and $\chi(M) \neq 0$, but I was not able to prove this.

Let $M$ be a compact oriented $n$-manifold. Denote $\Omega^k := \Lambda^k T^*M$ the vector bundle of differential $k$-forms, and let $\Omega^{\mathrm{odd}} := \oplus_{k\,\,\mathrm{odd}} \Omega^k$ and $\Omega^{\mathrm{even}} := \oplus_{k\,\,\mathrm{even}} \Omega^k$ be the bundles of odd and even differential forms, respectively. 

When are $\Omega^{\mathrm{odd}}$ and $\Omega^{\mathrm{even}}$ isomorphic as vector bundles over $M$?

Few observations:

• If $n$ is odd they are isomorphic, $\Omega^{\mathrm{odd}} \simeq \Omega^{\mathrm{even}}$, as can be seen e.g. by introducing a Riemannian metric $g$ on $M$, and observing that the Hodge star $\star: \Omega^k \to \Omega^{n - k}$ is an isomorphism for eack $k$ and hence also between $\Omega^{\mathrm{odd}}$ and $\Omega^{\mathrm{even}}$.

• More generally, if $\chi(M) = 0$, then $\Omega^{\mathrm{odd}} \simeq \Omega^{\mathrm{even}}$. Proof: there is a non-vanishing section $s$ of $\Omega^1 \simeq TM$. Write $\Omega^1 = \mathbb{R}s \oplus V$ for some vector bundle $V$. Then $$\Omega^{\mathrm{odd}} = s \wedge \Omega^{\mathrm{even}}(V) \oplus \Omega^{\mathrm{odd}}(V) \simeq \Omega(V) \simeq s \wedge \Omega^{\mathrm{odd}}(V) \oplus \Omega^{\mathrm{even}}(V) = \Omega^{\mathrm{even}},$$ where $\Omega(V)$ is the full exterior bundle of $V$, and $\Omega^{\mathrm{odd}/\mathrm{even}}(V)$ are the odd/even parts of $\Omega(V)$.

• However, if $n = 2$ they are not isomorphic unless $M$ is the torus: the Euler class of $\Omega^{\mathrm{odd}} = \Omega^1$ is then non-zero, while $\Omega^{\mathrm{even}} = \Omega^0 \oplus \Omega^2 \simeq \mathbb{R}^2$ has trivial Euler class.

• If $n = 2d$, $\Omega^{\mathrm{even}}$ and $\Omega^{\mathrm{odd}}$ can be seen to have identical Chern classes $c_i$, except possibly the top one $c_d$. (I define the Chern class of a real vector bundle to be the Chern class of its complexification.) Namely, if $SM \subset TM$ denotes the unit sphere bundle, and $\pi: SM \to M$ is the footpoint projection, then the pullbacks $\pi^*\Omega^{\mathrm{odd}} \simeq \pi^*\Omega^{\mathrm{even}}$ are isomorphic. (Proof: since there is a non-vanishing tautological section $\tau(x, v) := g_x(v, \bullet)$ of $\Omega^1$, we may repeat the argument in the second point.) By Gysin sequence, $\pi^*$ is an isomorphism on $H^\bullet(M)$ for $\bullet \leq n - 1$, so $c_i(\Omega^{\mathrm{odd}}) = c_i(\Omega^{\mathrm{even}})$ for $i \leq d - 1$, and $c_d(\Omega^{\mathrm{odd}}) - c_d(\Omega^{\mathrm{even}})$ is a multiple of $\chi(M)$ in $H^{2d}(M, \mathbb{Z})$.

• If $n = 4$, my computations suggest that $\Omega^{\mathrm{even}}$ and $\Omega^{\mathrm{odd}}$ have equal all Chern, and Stiefel-Whitney characteristic classes. One needs to use that $\Omega^2 = \Omega^1 \wedge \Omega^1$ and the formula for $c_2$ of a wedge product. In particular, $c_2(\Omega^{\mathrm{odd}}) - c_2(\Omega^{\mathrm{even}}) = 0$ is the $\it{zero\,\, multiple}$ of the Euler characteristic.

• My guess after all of this is that $\Omega^{\mathrm{even}} \not \simeq \Omega^{\mathrm{odd}}$ if $n$ is even and $\chi(M) \neq 0$, but I was not able to prove this.