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edited May 12 2011 at 21:09

Both questions are pretty comprehensively answered in the book by Buoncristiano, Rourke, and Sanderson, "A geometric approach to homology theory" (available at google books; start from Chapter 2). The answer is simpleas follows: the The pushforward $f_!:h^\ast(X)\to h^{\ast+n}(Y)$ in the cohomology theory $h^\ast$ (as well as the pullback in the dual homology theory) is defined whenever the map $f:X\to Y$ itself represents (in a sense to be discussed in a moment) a cohomology class $[f]\in h^n(Y)$in the cohomology theory $h^\ast$. . The definition of $f_!$ is simply by composition: $f_!([g])=[fg]$. (Here $n$ could be zero or negative (which is the only case I see considered elsewhere in this thread), but it could well can be positive. When $n$ is negative and the cohomology is ordinary, $[f]=0$ whenever $[f]$ is defined, but it is still meaningful to ask whether $[f]$ is defined for a given $f$. f$.)

Now the representatives of what do I mean by a representative? Well, an ordinary cohomology classes refers class in $H^1(X)\simeq [X,S^1]$, where $X$ is a polyhedron, can be represented (not to cochains or anything like that only by a cochain, but to the geometric construction also) by a transversal point-inverse $Z$ of a PL map $X\to S^1$, viewed as an embedding $Z\to X$. Two such representatives are equivalent if they cobound a transversal point-inverse of a PL homotopy $X\times I\to S^1$, viewed as a map $W\to X\times I$. (arbitrary) Here "transversal" means just that the point is taken in the interior of a $1$-simplex of a triangulation of $S^1$ which makes the map simplicial.) It turns out that this description of ordinary $1$-cohomology generalizes in an elegant way to arbitrary dimensions and arbitrary cohomology theoriesdeveloped , which was done in the B-R-S book. Other monographs based upon this geometric view of cohomology include Fenn's "Techniques of geometric topology" and Kreck's Differential Algebraic Topology and these may function as an introductory reading for the B-R-S book (in case you find it terse; another option is Rourke's ICM lecture).

Let me elaborate on the representatives

Here are some details. In the case of the unoriented PL cobordism theory , for instance (it is about the simplest theory from this geometric viewpoint), the representatives of cobordism classes are what B-R-S call "mock bundles"; I prefer calling them "comanifolds". An $n$-comanifold (where $n$ is any integer, positive or not) is a PL map of a polyhedron into a simplicial complex such that the preimage of every $i$-simplex is a PL $(i-n)$-manifold with boundary, where the and moreover its boundary equals the preimage of the boundary of the $i$-simplex. If you think about embedded $1$-comanifolds in a three-page book, for instance, you'll easily see that whenever one intersects the binding, it has to look locally like a triod orthogonal to the bindingin a neighborhood of that intersection. In fact, by a variant of the Pontryagin-Thom construction, a co-oriented embedded $1$-comanifold in a polyhedron $X$ is just a transversal point-inverse of a map $X\to S^1$. Of courseIn general, the representatives of co-oriented (not necessarily embedded) comanifolds represent oriented PL cobordism classesare co-oriented comanifolds.

The pushforward $f_!$ in ordinary integral cohomology is defined for every representative $f$ of an ordinary cohomology class, that is for a co-oriented comanifold with codimension two singularities. Note that these are a lot more general than bundles with a manifold fiber; in particular, every PL map between two PL manifolds (of either positive or negative codimension $n$) is included, after a small perturbation(this is proved in B-R-S). Admittedly, they do not these maps include not all maps whose homotopy fiber is homotopically a manifold, but in practice those can be often be replaced by reduced to bundles with a manifold fiber. The pushforward $f_!$ in stable cohomotopy is defined for $f$ a framed comanifold, etc. Etc. Everything generalizes straightforwardly to equivariant (including representation-graded) theories. Unfortunately the description of representatives is considerably a lot more complex in the case of $K$-theory (complex, say).

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edited May 12 2011 at 17:49

Sergey Melikhov
3,692●12●19

Both questions are pretty comprehensively answered in the book by Buoncristiano, Rourke, and Sanderson, "A geometric approach to homology theory" (available at google books; start from Chapter 2). The answer is simple: the pushforward $f_!:h^\ast(X)\to h^{\ast+n}(Y)$ in the cohomology theory $h^\ast$ (as well as the pullback in the dual homology theory) is defined whenever the map $f:X\to Y$ itself represents (in a sense to be discussed in a moment) a cohomology class $[f]\in h^n(Y)$ in the cohomology theory $h^\ast$. The "representatives" definition of $f_!$ is simply by composition: $f_!([g])=[fg]$. Here $n$ could be zero or negative (which is the only case I see considered elsewhere in this thread), but it could well be positive. When $n$ is negative and the cohomology is ordinary, $[f]=0$ whenever $[f]$ is defined, but it is still meaningful to ask whether $[f]$ is defined for a given $f$.

Now the representatives of cohomology classes refers not to cochains or anything like that but to the geometric construction of (arbitrary) cohomology theories developed in the B-R-S book. The definition of $f_!$ is simply by composition: $f_!([g])=[fg]$. Other monographs based upon this geometric view of cohomology include Fenn's "Techniques of geometric topology" and Kreck's Differential Algebraic Topology and these may function as an introductory reading for the B-R-S book (in case you find it terse; another option is Rourke's ICM lecture).

Let me elaborate on the representatives. In the case of the unoriented PL cobordism theory, for instance (it is about the simplest theory from this geometric viewpoint), the pushforward is defined for representatives of cobordism classes are what B-R-S call "mock bundles"; I prefer calling them "comanifolds". An $n$-comanifold (where $n$ is any integer, positive or not) is a PL map of a polyhedron into a simplicial complex such that the preimage of every $i$-simplex is a PL $(i-n)$-manifold with boundary, where the boundary equals the preimage of the boundary of the $i$-simplex. If you think about embedded $1$-comanifolds in a three-page book, for instance, you'll easily see that whenever one intersects the binding, it has to look locally like a triod orthogonal to the binding in a neighborhood of that intersection. In fact, a co-oriented embedded $1$-comanifold in a polyhedron $X$ is just a transversal point-inverse of a map $X\to S^1$. Of course, the pushforward in representatives of oriented PL cobordism is defined for classes are co-oriented comanifolds.

The pushforward $f_!$ in ordinary integral cohomology is defined for every representative $f$ of an ordinary cohomology class, that is for a co-oriented comanifolds comanifold with codimension two singularities. Note that these are a lot more general than bundles with a manifold fiber. They Admittedly, they do not include all maps whose homotopy fiber is homotopically a manifold, but in practice those can be often be replaced by bundles with a manifold fiber. The pushforward $f_!$ in stable cohomotopy is defined for $f$ a framed comanifoldscomanifold, etc. Everything generalizes straightforwardly to equivariant (including representation-graded) theories. Unfortunately the answer becomes description of representatives is considerably more complex in the case of $K$-theory (complex, say).

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edited May 12 2011 at 17:03

Sergey Melikhov
3,692●12●19

not only answering the question of for which maps f, but also answering the question of in which cohomology theories can we carry out such a construction

Both questions are pretty comprehensively answered in the book by Buoncristiano, Rourke, and Sanderson, "A geometric approach to homology theory" (available at google books; start from Chapter 2). The answer is simple: the pushforward $f_!:h^\ast(X)\to h^\ast(Y)$ h^{\ast+n}(Y)$ in the cohomology theory $h^\ast$ (as well as the pullback in the dual homology theory) is defined precisely for those maps whenever the map $f:X\to Y$ that are themselves representatives of itself represents a cohomology classes class $[f]\in h^\ast(Y)$ h^n(Y)$ in the cohomology theory $h^\ast$. The "representatives" refers not to cochains or anything like that but to the geometric construction of (arbitrary) cohomology theories developed in the B-R-S book. The definition of $f_!$ is simply by composition: $f_!([g])=[fg]$. Other monographs based upon this geometric view of cohomology include Fenn's "Techniques of geometric topology" and Kreck's Differential Algebraic Topology and these may function as an introductory reading for the B-R-S book (in case you find it terse; another option is Rourke's ICM lecture).

Let me elaborate. In the case of the unoriented PL cobordism theory, for instance (it is about the simplest theory from this geometric viewpoint), the pushforward is defined for what B-R-S call "mock bundles"; I prefer calling them "comanifolds". An $n$-comanifold (where $n$ is any integer, positive or not) is a PL map of a polyhedron into a simplicial complex such that the preimage of every $i$-simplex is a PL $(i-n)$-manifold with boundary, where the boundary equals the preimage of the boundary of the $i$-simplex. If you think about embedded $1$-comanifolds in a three-page book, for instance, you'll easily see that whenever one intersects the binding, it has to look locally like a triod orthogonal to the binding in a neighborhood of that intersection. In fact, a co-oriented embedded $1$-comanifold in a polyhedron $X$ is just a transversal point-inverse of a map $X\to S^1$. Of course, the pushforward in oriented PL cobordism is defined for co-oriented comanifolds.

The pushforward in ordinary integral cohomology is defined precisely for co-oriented comanifolds with codimension two singularities. Note that these are a lot more general than bundles with a manifold fiber. They do not include all maps whose homotopy fiber is homotopically a manifoldand such, but in practice those can be often be replaced by bundles with a manifold fiber. The pushforward in stable cohomotopy is defined precisely for framed comanifolds, etc. Everything generalizes straightforwardly to equivariant (including representation-graded) theories. Unfortunately the answer becomes considerably more complex in the case of $K$-theory (complex, say).

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edited May 12 2011 at 16:39

Sergey Melikhov
3,692●12●19

not only answering the question of for which maps f, but also answering the question of in which cohomology theories can we carry out such a construction

Both questions are pretty comprehensively answered in the book by Buoncristiano, Rourke, and Sanderson, "A geometric approach to homology theory" (available at google books; start from Chapter 2). The answer is simple: the pushforward $f_!:h^\ast(X)\to h^\ast(Y)$ in a the cohomology theory $h^\ast$ (as well as the pullback in the dual homology theory) is defined precisely for those maps $f:X\to Y$ that are themselves representatives of a cohomology class classes $[f]\in h^\ast(Y)$ in the cohomology theory . $h^\ast$. The "representation" representatives" refers not to cochains or anything like that but to the (beautiful) geometric construction of (arbitrary) cohomology theories developed in the B-R-S book. Other monographs based upon this geometric view of cohomology include Fenn's "Techniques of geometric topology" and Kreck's Differential Algebraic Topology and these may function as an introductory reading for the B-R-S book (in case you find it terse; another option is Rourke's ICM lecture).

In the case of the unoriented PL cobordism theory, for instance (it is about the simplest case theory from this geometric viewpoint), the pushforward is defined for what B-R-S call "mock bundles"; I prefer calling them "comanifolds". An $n$-comanifold (where $n$ is any integer, positive or not) is a PL map of a polyhedron into a simplicial complex such that the preimage of every $i$-simplex is a PL $(i-n)$-manifold with boundary, where the boundary equals the preimage of the boundary of the $i$-simplex. If you think about embedded $1$-comanifolds in a three-page book, for instance, you'll easily see that whenever one intersects the binding, it has to look locally like a triod orthogonal to the binding in a neighborhood of that intersection. In fact, a co-oriented embedded $1$-comanifold in a polyhedron $X$ is just a transversal point-inverse of a map $X\to S^1$. Of course, the pushforward in oriented PL cobordism is defined for co-oriented comanifolds.

The pushforward in ordinary integral cohomology is defined precisely for co-oriented comanifolds with codimension two singularities. Note that these are a lot more general than maps whose fiber is a manifold and such. The pushforward in stable cohomotopy is defined precisely for framed comanifolds, etc. Everything generalizes straightforwardly to equivariant (including representation-graded) theories. Unfortunately the answer becomes considerably more complex in the case of $K$-theory (complex, say).

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edited May 12 2011 at 16:19

Sergey Melikhov
3,692●12●19

not only answering the question of for which maps f, but also answering the question of in which cohomology theories can we carry out such a construction

Both questions are pretty comprehensively answered in the book by S. Buoncristiano, C. Rourke, and B. Sanderson, "A geometric approach to homology theory" (available at google books)books; start from Chapter 2). The answer is simple: the pushforward in a cohomology theory (as well as the pullback in the dual homology theory) are is defined precisely for those maps that are themselves represent representatives of a cohomology class in the cohomology theory. The "representation" refers not to cochains or anything like that but to the (beautiful) geometric construction of (arbitrary) cohomology theories developed in the B-R-S book. Other monographs based upon this geometric view of cohomology include Fenn's "Techniques of geometric topology" and Kreck's Differential Algebraic Topology.

In the case of the unoriented PL cobordism theory, for instance (it is about the simplest case from this geometric viewpoint), the pushforward is defined for what B-R-S call "mock bundles"; I've been I prefer calling them "comanifolds". An $n$-comanifold (where $n$ is any integer, positive or not) is a PL map of a polyhedron into the a simplicial complex such that the preimage of every $i$-simplex is a PL $(i-n)$-manifold with boundary, where the boundary equals the preimage of the boundary of the $i$-simplex. If you think about embedded $1$-comanifolds in a three-page book, for instance, you'll easily see that whenever one intersects the binding, it has to look locally like a triod orthogonal to the binding in a neighborhood of that intersection. In fact, a co-oriented embedded $1$-comanifold in a polyhedron $X$ is just a transversal point-inverse of a map $X\to S^1$. Of course, the pushforward in oriented PL cobordism is defined for co-oriented comanifolds.

The pushforward in ordinary integral cohomology is defined precisely for co-oriented comanifolds with codimension two singularities. These Note that these are surely a lot more general than maps whose fiber is a manifold and such. The pushforward in stable cohomotopy is defined precisely for framed comanifolds, etc. Everything generalizes straightforwardly to equivariant (including representation-graded) theories. Unfortunately , things get a lot the answer becomes considerably more involved with complex in the case of $K$-theory. K$-theory (complex, say).

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answered May 12 2011 at 13:44

Sergey Melikhov
3,692●12●19

not only answering the question of for which maps f, but also answering the question of in which cohomology theories can we carry out such a construction

Both questions are answered in the book by S. Buoncristiano, C. Rourke, and B. Sanderson, "A geometric approach to homology theory" (available at google books). The answer is simple: the pushforward in a cohomology theory (as well as the pullback in the dual homology theory) are defined precisely for those maps that themselves represent a cohomology class in the cohomology theory. The "representation" refers not to cochains or anything like that but to the (beautiful) geometric construction of cohomology theories developed in the book.

In the case of the unoriented cobordism theory, for instance, the pushforward is defined for what B-R-S call "mock bundles"; I've been calling them "comanifolds". An $n$-comanifold is a map into the simplicial complex such that the preimage of every $i$-simplex is a PL $(i-n)$-manifold with boundary, where the boundary equals the preimage of the boundary of the $i$-simplex. If you think about embedded $1$-comanifolds in a three-page book, for instance, you'll easily see that whenever one intersects the binding, it has to look locally like a triod orthogonal to the binding in a neighborhood of that intersection. In fact, a co-oriented embedded $1$-comanifold in a polyhedron $X$ is just a transversal point-inverse of a map $X\to S^1$. Of course, the pushforward in oriented cobordism is defined for co-oriented comanifolds.

The pushforward in ordinary integral cohomology is defined for co-oriented comanifolds with codimension two singularities. These are surely more general than maps whose fiber is a manifold and such. The pushforward in stable cohomotopy is defined for framed comanifolds, etc. Unfortunately, things get a lot more involved with complex $K$-theory.