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Suppose that $M^m$ is an $m$-dimensional, compact, connected manifold without boundary, $m\geq 2$, not necessarily orientable. $\newcommand{\bR}{\mathbb{R}}$ $\newcommand{\bZ}{\mathbb{Z}}$

To a smooth map $$ F: M\to\bR^m,\;\;F(p) =\big(\; f_1(p),\dotsc , f_m(p)\;\big) $$ we associate the top degree form $$ \omega_F=df_1\wedge \cdots\wedge df_m= F^* (du^1\wedge \cdots \wedge du^m), $$ where $u^1,\dotsc, u^m$ are the canonical Euclidean coordinates on $\bR^m$.

For generic $F$, the zero locus of $\omega_F$ is a hypersurface $W_F$ in $M$ and the homology class $[W_F]\in H_{m-1}(M,\bZ/2)$ is the Poincare dual of the Stieffel-Whitney class $w_1(TM)\in H^1(M,\bZ/2)$.

The noncompact manifold $M_F=M\setminus W_F$ is oriented by the top degree form $\omega_F$.

If $M$ where orientable, then an orientation $or_M$ on $M$ induces an orientation on $M_F$ and $$ \int_{(M_F,or_M)}\omega_F=\int_{(M,or_M)} \omega_F =\int_{(M,or_M)}d(f_1df_2\wedge\cdots\wedge df_m)= 0. $$ Suppose that $M$ is nonorientable. Is it true that for any orienation $or$ on $M_F$ we have $$ \int_{(M_F,or)} \omega_F \neq 0? $$ Comment Let us point out that if we denote by $or_F$ the orientation on $M_F$ induced by $\omega_F$, then $$ \int_{(M_F,or_F)} \omega_F >0, $$ for any $M$, orientable or not.

If $(M_i)_{1\leq i\leq k}$ are components of $M_F$, then for any orientation $or$ on $M_F$ there exist $\epsilon_i=\pm$ such that the orientation $or_i$ on $M_i$ induced by $or$ satisfies $or_i=\epsilon_i or_F$. Thus the question can be rephrased as follows.

If $M$ is non orientable is it true that, for any $\epsilon_i=\pm 1$, we have $$ \sum_{i=1}^k \epsilon_i \int_{(M_i,or_F)} \omega_F\neq 0? $$

Suppose that $M^m$ is an $m$-dimensional, compact, connected manifold without boundary, $m\geq 2$, not necessarily orientable. $\newcommand{\bR}{\mathbb{R}}$ $\newcommand{\bZ}{\mathbb{Z}}$

To a smooth map $$ F: M\to\bR^m,\;\;F(p) =\big(\; f_1(p),\dotsc , f_m(p)\;\big) $$ we associate the top degree form $$ \omega_F=df_1\wedge \cdots\wedge df_m= F^* (du^1\wedge \cdots \wedge du^m), $$ where $u^1,\dotsc, u^m$ are the canonical Euclidean coordinates on $\bR^m$.

For generic $F$, the zero locus of $\omega_F$ is a hypersurface $W_F$ in $M$ and the homology class $[W_F]\in H_{m-1}(M,\bZ/2)$ is the Poincare dual of the Stieffel-Whitney class $w_1(TM)\in H^1(M,\bZ/2)$.

The noncompact manifold $M_F=M\setminus W_F$ is oriented by the top degree form $\omega_F$.

If $M$ where orientable, then an orientation $or_M$ on $M$ induces an orientation on $M_F$ and $$ \int_{(M_F,or_M)}\omega_F=\int_{(M,or_M)} \omega_F =\int_{(M,or_M)}d(f_1df_2\wedge\cdots\wedge df_m)= 0. $$ Suppose that $M$ is nonorientable. Is it true that for any orientation $or$ on $M_F$ we have $$ \int_{(M_F,or)} \omega_F \neq 0? $$ Comment. In the initial form the question was ill posed as indicated by Robert Bryant and I modified it. Here is an alternate reformulation.

Let us point out that if we denote by $or_F$ the orientation on $M_F$ induced by $\omega_F$, then $$ \int_{(M_F,or_F)} \omega_F >0, $$ for any $M$, orientable or not.

If $(M_i)_{1\leq i\leq k}$ are components of $M_F$, then for any orientation $or$ on $M_F$ there exist $\epsilon_i=\pm$ such that the orientation $or_i$ on $M_i$ induced by $or$ satisfies $or_i=\epsilon_i or_F$. Thus the question can be rephrased as follows.

If $M$ is non orientable is it true that, for any $\epsilon_i=\pm 1$, we have $$ \sum_{i=1}^k \epsilon_i \int_{(M_i,or_F)} \omega_F\neq 0? $$

Suppose that $M^m$ is an $m$-dimensional, compact, connected manifold without boundary, $m\geq 2$, not necessarily orientable. $\newcommand{\bR}{\mathbb{R}}$ $\newcommand{\bZ}{\mathbb{Z}}$

To a smooth map $$ F: M\to\bR^m,\;\;F(p) =\big(\; f_1(p),\dotsc , f_m(p)\;\big) $$ we associate the top degree form $$ \omega_F=df_1\wedge \cdots\wedge df_m= F^* (du^1\wedge \cdots \wedge du^m), $$ where $u^1,\dotsc, u^m$ are the canonical Euclidean coordinates on $\bR^m$.

For generic $F$, the zero locus of $\omega_F$ is a hypersurface $W_F$ in $M$ and the homology class $[W_F]\in H_{m-1}(M,\bZ/2)$ is the Poincare dual of the Stieffel-Whitney class $w_1(TM)\in H^1(M,\bZ/2)$.

The noncompact manifold $M_F=M\setminus W_F$ is oriented by the top degree form $\omega_F$.

If $M$ where orientable, then an orientation $or_M$ on $M$ induces an orientation on $M_F$ and $$ \int_{(M_F,or_M)}\omega_F=\int_{(M,or+M)} \omega_F =\int_{(M,or_M)}d(f_1df_2\wedge\cdots\wedge df_m)= 0. $$ Suppose that $M$ is nonorientable. Is it true that for any orienation $or$ on $M_F$ we have $$ \int_{(M_F,or)} \omega_F \neq 0? $$ Comment Let us point out that if we denote by $or_F$ the orientation on $M_F$ induced by $\omega_F$, then $$ \int_{(M_F,or_F)} \omega_F >0, $$ for any $M$, orientable or not.

Suppose that $M^m$ is an $m$-dimensional, compact, connected manifold without boundary, $m\geq 2$, not necessarily orientable. $\newcommand{\bR}{\mathbb{R}}$ $\newcommand{\bZ}{\mathbb{Z}}$

To a smooth map $$ F: M\to\bR^m,\;\;F(p) =\big(\; f_1(p),\dotsc , f_m(p)\;\big) $$ we associate the top degree form $$ \omega_F=df_1\wedge \cdots\wedge df_m= F^* (du^1\wedge \cdots \wedge du^m), $$ where $u^1,\dotsc, u^m$ are the canonical Euclidean coordinates on $\bR^m$.

For generic $F$, the zero locus of $\omega_F$ is a hypersurface $W_F$ in $M$ and the homology class $[W_F]\in H_{m-1}(M,\bZ/2)$ is the Poincare dual of the Stieffel-Whitney class $w_1(TM)\in H^1(M,\bZ/2)$.

The noncompact manifold $M_F=M\setminus W_F$ is oriented by the top degree form $\omega_F$.

If $M$ where orientable, then an orientation $or_M$ on $M$ induces an orientation on $M_F$ and $$ \int_{(M_F,or_M)}\omega_F=\int_{(M,or_M)} \omega_F =\int_{(M,or_M)}d(f_1df_2\wedge\cdots\wedge df_m)= 0. $$ Suppose that $M$ is nonorientable. Is it true that for any orienation $or$ on $M_F$ we have $$ \int_{(M_F,or)} \omega_F \neq 0? $$ Comment Let us point out that if we denote by $or_F$ the orientation on $M_F$ induced by $\omega_F$, then $$ \int_{(M_F,or_F)} \omega_F >0, $$ for any $M$, orientable or not.

If $(M_i)_{1\leq i\leq k}$ are components of $M_F$, then for any orientation $or$ on $M_F$ there exist $\epsilon_i=\pm$ such that the orientation $or_i$ on $M_i$ induced by $or$ satisfies $or_i=\epsilon_i or_F$. Thus the question can be rephrased as follows.

If $M$ is non orientable is it true that, for any $\epsilon_i=\pm 1$, we have $$ \sum_{i=1}^k \epsilon_i \int_{(M_i,or_F)} \omega_F\neq 0? $$

Suppose that $M^m$ is an $m$-dimensional, compact, connected manifold without boundary, $m\geq 2$, not necessarily orientable. $\newcommand{\bR}{\mathbb{R}}$ $\newcommand{\bZ}{\mathbb{Z}}$

To a smooth map $$ F: M\to\bR^m,\;\;F(p) =\big(\; f_1(p),\dotsc , f_m(p)\;\big) $$ we associate the top degree form $$ \omega_F=df_1\wedge \cdots\wedge df_m= F^* (du^1\wedge \cdots \wedge du^m), $$ where $u^1,\dotsc, u^m$ are the canonical Euclidean coordinates on $\bR^m$.

For generic $F$, the zero locus of $\omega_F$ is a hypersurface $W_F$ in $M$ and the homology class $[W_F]\in H_{m-1}(M,\bZ/2)$ is the Poincare dual of the Stieffel-Whitney class $w_1(TM)\in H^1(M,\bZ/2)$.

The noncompact manifold $M_F=M\setminus W_F$ is oriented by the top degree form $\omega_F$.

If $M$ where orientable, then $$ \int_{M_F}\omega_F=\int_M \omega_F =\int_M d(f_1df_2\wedge\cdots\wedge df_m)= 0. $$ Is it true that $$ \int_{M_F} \omega_F \neq 0 $$ if $M$ is nonorientable?

Suppose that $M^m$ is an $m$-dimensional, compact, connected manifold without boundary, $m\geq 2$, not necessarily orientable. $\newcommand{\bR}{\mathbb{R}}$ $\newcommand{\bZ}{\mathbb{Z}}$

To a smooth map $$ F: M\to\bR^m,\;\;F(p) =\big(\; f_1(p),\dotsc , f_m(p)\;\big) $$ we associate the top degree form $$ \omega_F=df_1\wedge \cdots\wedge df_m= F^* (du^1\wedge \cdots \wedge du^m), $$ where $u^1,\dotsc, u^m$ are the canonical Euclidean coordinates on $\bR^m$.

For generic $F$, the zero locus of $\omega_F$ is a hypersurface $W_F$ in $M$ and the homology class $[W_F]\in H_{m-1}(M,\bZ/2)$ is the Poincare dual of the Stieffel-Whitney class $w_1(TM)\in H^1(M,\bZ/2)$.

The noncompact manifold $M_F=M\setminus W_F$ is oriented by the top degree form $\omega_F$.

If $M$ where orientable, then an orientation $or_M$ on $M$ induces an orientation on $M_F$ and $$ \int_{(M_F,or_M)}\omega_F=\int_{(M,or+M)} \omega_F =\int_{(M,or_M)}d(f_1df_2\wedge\cdots\wedge df_m)= 0. $$ Suppose that $M$ is nonorientable. Is it true that for any orienation $or$ on $M_F$ we have $$ \int_{(M_F,or)} \omega_F \neq 0? $$ Comment Let us point out that if we denote by $or_F$ the orientation on $M_F$ induced by $\omega_F$, then $$ \int_{(M_F,or_F)} \omega_F >0, $$ for any $M$, orientable or not.