promted by discussion in another thread (here) I have reworked the Idea-section of this entry here:

Tried to clarify

the crucial distinction between classifying spaces and moduli stacks (“universes”).

(In particular I shrunk the out-of-place paragraph on the Grothendieck construction to a small side-remark – but this could well be deleted a altogether.)

the fact that there are classifying spaces of other things than principal 1-bundles, for instance of $n$-bundles and of cohomology theories (mentioned EM-spaces as classifying spaces for ordinary cohomology and mentioned the Brown representability theorem). One could expand on this point much more, of course.

Thank you! I will collect the pages edited to add a link here:

Multiple spaces: Introduction to Cobordism and Complex Oriented Cohomology, Landweber exact functor theorem, Pontrjagin class, Thom’s theorem, basic ideas of moduli stacks of curves and Gromov-Witten theory, classifying space, group cohomology, homology of MG, orthogonal group, special orthogonal group, stable unitary group, string structure, tangential structure, topological K-theory, twisted smooth cohomology in string theory, universal vector bundle

$B O(n)$ only: Thom space, Thom spectrum, real projective space

$B SO(n)$ only: Pontryagin’s theorem, SO(3), SO(4), SO(8), Sullivan model of a spherical fibration

$B U(n)$ only: Atiyah-Segal completion theorem, Chern class, Conner-Floyd Chern class, Pontrjagin ring, Milnor-Quillen theorem on MU, Snaith theorem, Young diagram, algebraic cobordism, basic complex line bundle on the 2-sphere, complex oriented cohomology theory, complex projective space, differential function complex, flux quantization in superconductors – section, formal group, geometry of physics – flux quantization, geometry of physics – fundamental super p-branes, infinite-dimensional sphere, infinity-Chern-Weil theory introduction, line bundle, motivation for sheaves, cohomology and higher stacks, real oriented cohomology theory, unitary group, universal complex line bundle, universal complex orientation on MU, zero-section into Thom space of universal line bundle is weak equivalence

$B SU(n)$ only: SU2-instantons from the correct maths to the traditional physics story, cohomotopy

I intend to publish the pages for BO(n), BSO(n), BU(n), BSU(n) within this week.

]]>Regarding your first question:

the code

```
[[BO(n)|$B O(n)$]]
```

works fine for me, I don’t get a spurious whitespace. Do you mean something else?

$\,$

Regarding your second question:

No, minor edits do not need to be announced. I don’t announce many little edits, in order not to flood the nForum (even more than I already do).

When making related edits on a whole range of pages, I usually announce this for only one of them, indicating in the comment which other pages are being edited similarly.

]]>Linked future pages for BO(n) and BU(n), the classifying spaces of the orthogonal group $O(n)$ and unitary group $U(n)$ respectively. (See discussion on Stiefel-Whitney class.)

I have two questions here: How are formulas like $B O(n)$ and $B U(n)$ supposed to be linked? By removing the formula environment or not? I did the former as otherweise there would have to be an empty space after “B”. I’ve also collected a list of pages, on which the new pages for $B O(n)$, $B S O(n)$, $B U (n)$ and $B S U(n)$ have to be linked. I would do this before creating them. Do I need to write a message for the forum for such a small edit? When having done so for Field, it kinda spammed the forum full with those messages. If it’s necessary, I would spread the edits over a week to avoid this.

]]>I have added more explicity statement of the Milnor classifying theorem, emphasizing that it works for any Hausdorff structure group and over any paracompact Hausdorff space.

I am wondering about the following, which sounds trivial, but is somewhat subtle:

In which generality are diffeological Cech 1-cocycles relative to a good open cover of a Cartesian space $\mathbb{R}^n$ isomorphic to the trivial cocycle?

The classical Milnor classification theorem implies this for the case that the structure group is a D-topological Hausdorff group. Does it actually fail for non-Hausdorff D-topological groups? How about general diffeological groups?

One should use a method that concretely trivializes the Cech cocycle without arguing indireclt via its classifying space. But constructions of such trivializations that spring to mind turn out to implicitly depend on the fact to be proven.

]]>added pointer to:

- Loring Tu, Section I.5 in:
*Introductory Lectures on Equivariant Cohomology*, Annals of Mathematics Studies**204**, AMS 2020 (ISBN:9780691191744)

added pointer to:

- Richard Palais, Section 2.6 of:
*The classification of $G$-spaces*, Memoirs of the AMS**36**, 1960 (ISBN:978-0-8218-9979-3 pdf)

[ obsolete ]

]]>added pointer to:

- Michael C. McCord,
*Classifying Spaces and Infinite Symmetric Products*, Transactions of the American Mathematical Society, Vol. 146 (Dec., 1969), pp. 273-298 (jstor:1995173, pdf)

Added pointer to:

- Gerd Rudolph, Matthias Schmidt, Thm. 3.5.1 of:
*Differential Geometry and Mathematical Physics Part II. Fibre Bundles, Topology and Gauge Fields*, Springer 2017 (doi:10.1007/978-94-024-0959-8)

added more of the classical textbook references:

Dale Husemoeller, Section 4.12, 4.13 of:

*Fibre bundles*, McGraw-Hill 1966 (300 p.); Springer Graduate Texts in Math.**20**, 2nd ed. 1975 (327 p.), 3rd. ed. 1994 (353 p.) (gBooks, pdf)Norman Steenrod, section II.19 of:

*The topology of fibre bundles*, Princeton Mathematical Series 14, Princeton Univ. Press, 1951 (jstor:j.ctt1bpm9t5)

Where the References-section started out with “Original articles…” I added “… in topological homotopy theory” and then further down I added “Discussion in simplicial homotopy theory” with all these pointers:

Paul Goerss, Rick Jardine, Section V.3 of:

*Simplicial homotopy theory*, Progress in Mathematics, Birkhäuser (1996) (doi:10.1007/978-3-0346-0189-4))David Roberts, Danny Stevenson,

*Simplicial principal bundles in parametrized spaces*, New York Journal of Mathematics Volume 22 (2016) 405-440 (arXiv:1203.2460)Danny Stevenson,

*Classifying theory for simplicial parametrized groups*(arXiv:1203.2461)Thomas Nikolaus, Urs Schreiber, Danny Stevenson,

*Principal ∞-bundles – Presentations*, Journal of Homotopy and Related Structures, Volume 10, Issue 3 (2015), pages 565-622 (doi:10.1007/s40062-014-0077-4, arXiv:1207.0249)

added a sub-section “References – For equivariant bundles” (here)

]]>There is a mention in a Seminar of Moore in Numdam. I have not checked if it does the construction.

]]>have added these pointers:

John Milnor,

*Construction of Universal Bundles, II*, Annals of Mathematics Second Series, Vol. 63, No. 3 (May, 1956), pp. 430-436 (jstor:1970012)Graeme Segal,

*Classifying spaces and spectral sequences*, Publications Mathématiques de l’IHÉS, Volume 34 (1968), p. 105-112 (numdam:PMIHES_1968__34__105_0)Norman Steenrod,

*Milgram’s classifying space of a topological group*, Topology Volume 7, Issue 4, November 1968, Pages 349-368 (doi:10.1016/0040-9383(68)90012-8Jim Stasheff,

*H-spaces and classifying spaces: foundations and recent developments*, Algebraic topology (Proc. Sympos. Pure Math., Vol. XXII, Univ. Wisconsin, Madison, Wis., 1970), Providence, R.I.: American Mathematical Society (1971) pp. 247–272

Thanks! Am adding these…

]]>Maybe for more information:

- Stasheff, James D. (1971), “H-spaces and classifying spaces: foundations and recent developments”, Algebraic topology (Proc. Sympos. Pure Math., Vol. XXII, Univ. Wisconsin, Madison, Wis., 1970), Providence, R.I.: American Mathematical Society, pp. 247–272

This opens with

The theory of classifying spaces for principal bundles has a long history in topology [Mi,Se, St]

- [MI] J. Milnor, Construction of universal bundles. II., Ann. of Math. (2)63(1956), 430–436.
- [Se]G. Segal,Classifying spaces and spectral sequences, Inst. Hautes ́Etudes Sci. Publ. Math. No. 34(1968), 105–112.
- [St]N.E. Steenrod,Milgram’s classifying space of a topological group, Topology7(1968), 349–368

What’s an original reference for the construction of classifying spaces (via Grassmannian’s etc)?

The entry currently offers

- Henri Cartan, Laurent Schwartz,
*Le théoréme d’Atiyah-Singer*Séminaire 1963/1964. New York: Benjamin 1967.

but I can find no electronic copy of a reference of precisely this title, and in those of similar title I don’t see the definition of classifying spaces.

(Maybe it was me who added that pointer, but I forget.)

What is an(other) record of the original conception of classifying spaces of G-principal bundles?

]]>Corrected misspelling of Hatcher’s name.

Anonymous

]]>I have now added in also all the remaining statements and proofs in the unitary case at *Classifying space – Examples – For orthogonal and unitary principal bundles*.

I think I’ve read that the attribution to Yoneda of this famous lemma is slightly complicated, since he seems not to have actually published a proof himself; see here. In any case, while I don’t know the answer to Zoran’s question, I shouldn’t be surprised if he did know about classifying spaces, since his name is also given to certain topics in homological algebra (I’m thinking for example of $Ext^n(A, B)$ defined by classes of long exact extensions starting with $B$ and ending with $A$), which have close connections to classifying spaces and bar constructions and the like.

]]>Was Yoneda aware of the idea of classifying spaces when proving the Yoneda lemma ? (I guess that the Grothendieck, who done it independenty, was aware of).

]]>I have added a bunch of the classical material, statements and proofs, to *Classifying space – Examples – For orthogonal and unitary principal bundles*.