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One place where negative dimensional manifolds appear naturally is complex cobordism $U^*$. Intuitively, elements of the abelian group $U^n(X)$ are represented by families of $(-n)$-dimensional manifolds varying over $X$.

Let us only define $U^*(X)$ for $X$ a finite-dimensional manifold, without boundary but not necessarily compact. Suppose $X$ is one such, of dimension $d$. Then, the abelian groups $U^n(X)$ are zero for $n>d$ and can be nonzero for all $n\leqslant d$, also negative. It is generated by pairs $(\iota:M\hookrightarrow E,c)$ where $M$ is a $d-n$-manifold, $E$ is either the total space of a complex vector bundle over $X$ (for $n$ even) or $\mathbb R$ times such total space (for $n$ odd), and $c$ is a complex structure on the normal bundle of $\iota$ which is an embedding. Relations identify pairs that are cobordant. For details, see

Quillen, D., Elementary proofs of some results of cobordism theory using Steenrod operations, Adv. Math. 7, 29-56 (1971). ZBL0214.50502.

For an element of $U^{-n}(X)$ represented by $(\iota,c)$ as above the composite of the projection of the bundle with $\iota$ gives a map $M\to X$ from a $d+n$-manifold $M$ to the $d$-manifold $X$ which can be thought of as a family of $n$-manifolds varying over $X$. This is intuitively understandable for $n\geqslant0$, but also keeps making sense for $n\leqslant0$ all the way until $n=-d$.

One place where negative dimensional manifolds appear naturally is complex cobordism $U^*$. Intuitively, elements of the abelian group $U^n(X)$ are represented by families of $(-n)$-dimensional manifolds varying over $X$.

Let us only define $U^*(X)$ for $X$ a finite-dimensional manifold, without boundary but not necessarily compact. Suppose $X$ is one such, of dimension $d$. Then, the abelian group $U^n(X)$ is zero for $n>d$ and can be nonzero for all $n\leqslant d$, also negative. It is generated by pairs $(\iota:M\hookrightarrow E,c)$ where $M$ is a $d-n$-manifold, $E$ is either the total space of a complex vector bundle over $X$ (for $n$ even) or $\mathbb R$ times such total space (for $n$ odd), and $c$ is a complex structure on the normal bundle of $\iota$ which is an embedding. Relations identify pairs that are cobordant. For details, see

Quillen, D., Elementary proofs of some results of cobordism theory using Steenrod operations, Adv. Math. 7, 29-56 (1971). ZBL0214.50502.

For an element of $U^{-n}(X)$ represented by $(\iota,c)$ as above the composite of the projection of the bundle with $\iota$ gives a map $M\to X$ from a $d+n$-manifold $M$ to the $d$-manifold $X$ which can be thought of as a family of $n$-manifolds varying over $X$. This is intuitively understandable for $n\geqslant0$, but also keeps making sense for $n\leqslant0$ all the way until $n=-d$.

One place where negative dimensional manifolds appear naturally is complex cobordism $U^*$. Intuitively, elements of the abelian group $U^n(X)$ are represented by families of $(-n)$-dimensional manifolds varying over $X$.

Let us only define $U^*(X)$ for $X$ a finite-dimensional manifold, without boundary but not necessarily compact. Suppose $X$ is one such, of dimension $d$. Then, the abelian groups $U^n(X)$ are zero for $n>d$ and can be nonzero for all $n\leqslant d$, also negative. The group $U^n(X)$ is then zero for $n>d$ and can be nonzero for any $n\leqslant d$, also negative. It is generated by pairs $(\iota:M\hookrightarrow E,c)$ where $M$ is a $d-n$-manifold, $E$ is either the total space of a complex vector bundle over $X$ (for $n$ even) or $\mathbb R$ times such total space (for $n$ odd), and $c$ is a complex structure on the normal bundle of $\iota$ which is an embedding. Relations identify pairs that are cobordant. For details, see

Quillen, D., Elementary proofs of some results of cobordism theory using Steenrod operations, Adv. Math. 7, 29-56 (1971). ZBL0214.50502.

For an element of $U^{-n}(X)$ represented by $(\iota,c)$ as above the composite of the projection of the bundle with $\iota$ gives a map $M\to X$ from a $d+n$-manifold $M$ to the $d$-manifold $X$ which can be thought of as a family of $n$-manifolds varying over $X$. This is intuitively understandable for $n\geqslant0$, but also keeps making sense for $n\leqslant0$ all the way until $n=-d$.

One place where negative dimensional manifolds appear naturally is complex cobordism $U^*$. Intuitively, elements of the abelian group $U^n(X)$ are represented by families of $(-n)$-dimensional manifolds varying over $X$.

Let us only define $U^*(X)$ for $X$ a finite-dimensional manifold, without boundary but not necessarily compact. Suppose $X$ is one such, of dimension $d$. Then, the abelian groups $U^n(X)$ are zero for $n>d$ and can be nonzero for all $n\leqslant d$, also negative. It is generated by pairs $(\iota:M\hookrightarrow E,c)$ where $M$ is a $d-n$-manifold, $E$ is either the total space of a complex vector bundle over $X$ (for $n$ even) or $\mathbb R$ times such total space (for $n$ odd), and $c$ is a complex structure on the normal bundle of $\iota$ which is an embedding. Relations identify pairs that are cobordant. For details, see

Quillen, D., Elementary proofs of some results of cobordism theory using Steenrod operations, Adv. Math. 7, 29-56 (1971). ZBL0214.50502.

For an element of $U^{-n}(X)$ represented by $(\iota,c)$ as above the composite of the projection of the bundle with $\iota$ gives a map $M\to X$ from a $d+n$-manifold $M$ to the $d$-manifold $X$ which can be thought of as a family of $n$-manifolds varying over $X$. This is intuitively understandable for $n\geqslant0$, but also keeps making sense for $n\leqslant0$ all the way until $n=-d$.

One place where negative dimensional manifolds appear naturally is complex cobordism $U^*$. Intuitively, elements of the abelian group $U^n(X)$ are represented by families of $(-n)$-dimensional manifolds varying over $X$.

Let us only define $U^*(X)$ for $X$ a finite-dimensional manifold, without boundary but not necessarily compact. Suppose $X$ is one such, of dimension $d$. Then, the abelian groups $U^n(X)$ are zero for $n>d$ and can be nonzero for all $n\leqslant d$, also negative. The group $U^n(X)$ is then zero for $n>d$ and can be nonzero for any $n\leqslant d$, also negative. It is generated by pairs $(\iota:M\hookrightarrow E,c)$ where $M$ is a $d-n$-manifold, $E$ is either the total space of a complex vector bundle over $X$ (for $n$ even) or $\mathbb R$ times such total space (for $n$ odd), and $c$ is a complex structure on the normal bundle of $\iota$ which is an embedding. Relations identify pairs that are cobordant. For details, see

Quillen, D., Elementary proofs of some results of cobordism theory using Steenrod operations, Adv. Math. 7, 29-56 (1971). ZBL0214.50502.

For an element of $U^{-n}(X)$ represented by $(\iota,c)$ as above the composite of the projection of the bundle with $\iota$ gives a map $M\to X$ from an $d+n$-manifold $M$ to the $d$-manifold $X$ which can be thought of as a family of $n$-manifolds varying over $X$. This is intuitively understandable for $n\geqslant0$, but also keeps making sense for $n\leqslant0$ all the way until $n=-d$.

One place where negative dimensional manifolds appear naturally is complex cobordism $U^*$. Intuitively, elements of the abelian group $U^n(X)$ are represented by families of $(-n)$-dimensional manifolds varying over $X$.

Let us only define $U^*(X)$ for $X$ a finite-dimensional manifold, without boundary but not necessarily compact. Suppose $X$ is one such, of dimension $d$. Then, the abelian groups $U^n(X)$ are zero for $n>d$ and can be nonzero for all $n\leqslant d$, also negative. The group $U^n(X)$ is then zero for $n>d$ and can be nonzero for any $n\leqslant d$, also negative. It is generated by pairs $(\iota:M\hookrightarrow E,c)$ where $M$ is a $d-n$-manifold, $E$ is either the total space of a complex vector bundle over $X$ (for $n$ even) or $\mathbb R$ times such total space (for $n$ odd), and $c$ is a complex structure on the normal bundle of $\iota$ which is an embedding. Relations identify pairs that are cobordant. For details, see

Quillen, D., Elementary proofs of some results of cobordism theory using Steenrod operations, Adv. Math. 7, 29-56 (1971). ZBL0214.50502.

For an element of $U^{-n}(X)$ represented by $(\iota,c)$ as above the composite of the projection of the bundle with $\iota$ gives a map $M\to X$ from a $d+n$-manifold $M$ to the $d$-manifold $X$ which can be thought of as a family of $n$-manifolds varying over $X$. This is intuitively understandable for $n\geqslant0$, but also keeps making sense for $n\leqslant0$ all the way until $n=-d$.