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tidied up notations, added tags
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edit approved Apr 1, 2016 at 12:00
Najib Idrissi
edit approved Apr 1, 2016 at 12:00
Najib Idrissi
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Source request for $H*$H^*(BTOPB\mathrm{TOP},\mathbb{Q})\cong H* \cong H^*(BO,\mathbb{Q})$

Let $BTOP$$B\mathrm{TOP}$ denote the classifying space for microbundles, i.e. $BHomeo(\mathbb{R}^n,0)$$B\operatorname{Homeo}(\mathbb{R}^n,0)$. Now we get a map from $BO$ to $BTOP$$B\mathrm{TOP}$ via the inclusion. Let $f$ denote the corresponding map in the rational cohomology rings.

Andrew Ranicki claims in his paper "On the construction and topological invariance of the Pontryagin classes" that the surjectivity of $f$ is equivalent to the topological invariance of the Pontryagin numbers, which is fine. But then he also writes without a source, that it is now known, that these two rings are actually isomorphic.

So on the one hand I would like to get a source for this "fact" on the other hand I would like to know if there is some nice meaning behind the injectivity of this map, like for surjectivity. I mean there is the obvious translation, that whenever to rational characteristic classes for micro bundles agree on vector bundles then they have to be the same, but maybe there is a stronger statement.

Link to the paper: http://www.maths.ed.ac.uk/~aar/papers/invtop.pdf p.311

Source request for $H*(BTOP,\mathbb{Q})\cong H*(BO,\mathbb{Q})$

Let $BTOP$ denote the classifying space for microbundles, i.e. $BHomeo(\mathbb{R}^n,0)$. Now we get a map from $BO$ to $BTOP$ via the inclusion. Let $f$ denote the corresponding map in the rational cohomology rings.

Andrew Ranicki claims in his paper "On the construction and topological invariance of the Pontryagin classes" that the surjectivity of $f$ is equivalent to the topological invariance of the Pontryagin numbers, which is fine. But then he also writes without a source, that it is now known, that these two rings are actually isomorphic.

So on the one hand I would like to get a source for this "fact" on the other hand I would like to know if there is some nice meaning behind the injectivity of this map, like for surjectivity. I mean there is the obvious translation, that whenever to rational characteristic classes for micro bundles agree on vector bundles then they have to be the same, but maybe there is a stronger statement.

Link to the paper: http://www.maths.ed.ac.uk/~aar/papers/invtop.pdf p.311

Source request for $H^*(B\mathrm{TOP},\mathbb{Q}) \cong H^*(BO,\mathbb{Q})$

Let $B\mathrm{TOP}$ denote the classifying space for microbundles, i.e. $B\operatorname{Homeo}(\mathbb{R}^n,0)$. Now we get a map from $BO$ to $B\mathrm{TOP}$ via the inclusion. Let $f$ denote the corresponding map in the rational cohomology rings.

Andrew Ranicki claims in his paper "On the construction and topological invariance of the Pontryagin classes" that the surjectivity of $f$ is equivalent to the topological invariance of the Pontryagin numbers, which is fine. But then he also writes without a source, that it is now known, that these two rings are actually isomorphic.

So on the one hand I would like to get a source for this "fact" on the other hand I would like to know if there is some nice meaning behind the injectivity of this map, like for surjectivity. I mean there is the obvious translation, that whenever to rational characteristic classes for micro bundles agree on vector bundles then they have to be the same, but maybe there is a stronger statement.

Link to the paper: http://www.maths.ed.ac.uk/~aar/papers/invtop.pdf p.311

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ThorbenK
ThorbenK
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Source request for $H*(BTOP,\mathbb{Q})\cong H*(BO,\mathbb{Q})$

Let $BTOP$ denote the classifying space for microbundles, i.e. $BHomeo(\mathbb{R}^n,0)$. Now we get a map from $BO$ to $BTOP$ via the inclusion. Let $f$ denote the corresponding map in the rational cohomology rings.

Andrew Ranicki claims in his paper "On the construction and topological invariance of the Pontryagin classes" that the surjectivity of $f$ is equivalent to the topological invariance of the Pontryagin numbers, which is fine. But then he also writes without a source, that it is now known, that these two rings are actually isomorphic.

So on the one hand I would like to get a source for this "fact" on the other hand I would like to know if there is some nice meaning behind the injectivity of this map, like for surjectivity. I mean there is the obvious translation, that whenever to rational characteristic classes for micro bundles agree on vector bundles then they have to be the same, but maybe there is a stronger statement.

Link to the paper: http://www.maths.ed.ac.uk/~aar/papers/invtop.pdf p.311