寻求分类空间及其上同调的参考文献，以理解示性类

Hey there! It sounds like you're on a great path to connecting classifying spaces, their cohomology groups, and characteristic classes—Husemoller's Fiber Bundles is a solid start, but I totally get why the sparse classifying space coverage and Milnor's tricky construction might leave you wanting more. Let's break this down with some go-to references and a concise overview of the key ideas.

Recommended References #

Here are some books that fill the gaps and approach the topic from angles that complement Husemoller:

• Allen Hatcher's Vector Bundles and K-Theory: This is perfect for building intuition. Hatcher explains classifying spaces with a heavy geometric focus, walks through Milnor's construction step-by-step with clear motivation, and ties everything directly to vector bundles and their invariants. The style is accessible but rigorous, and it’s great for bridging the gap between basic fiber bundle theory and characteristic classes.
• Bott & Tu's Differential Forms in Algebraic Topology: While centered on differential forms, this book masterfully links de Rham cohomology to characteristic classes and classifying spaces. Their treatment of Chern classes and Stiefel-Whitney classes via the universal bundles on Grassmannians (the concrete classifying spaces for classical groups) is incredibly illuminating—you’ll see exactly how cohomology classes translate to bundle invariants.
• Milnor & Stasheff's Characteristic Classes: The classic text on the subject. It dives deep into classifying spaces, with systematic coverage of the Milnor construction and the cohomology of (BG) for classical Lie groups. The book is packed with examples and computational details that will help you map cohomology groups directly to characteristic class properties (like Whitney products or naturality). Pair this with Hatcher if Milnor’s writing feels too dense at first.
• Tammo tom Dieck's Algebraic Topology: If you want to solidify the topological group foundations behind classifying spaces, this book has an exhaustive, well-motivated section on Milnor’s construction and the homotopy-theoretic properties of (BG). It connects the abstract classification theorem to concrete topological constructions, which should clarify the "why" behind Milnor’s approach.

Concise Overview of Classifying Spaces #

Let’s distill the key ideas for topological groups, Lie groups, and vector bundles:

Topological Groups & Their Classifying Spaces #

For a topological group (G), a classifying space (BG) is a space with a universal property:

Every (G)-principal bundle over a space (X) is isomorphic to the pullback of the universal principal bundle (EG \to BG) via some continuous map (f: X \to BG). Moreover, two bundles are isomorphic if and only if their corresponding maps are homotopic.

Milnor’s construction of (BG) is all about building this "universal" space:

• Start with an infinite product of copies of (G), then define an equivalence relation that glues adjacent copies using (G)-multiplication. The result is (EG)—a contractible space where (G) acts freely.
• (BG) is then the quotient (EG/G). Since (EG) is contractible, it only carries the trivial (G)-bundle, so every possible (G)-bundle arises as a pullback from (BG).

Lie Groups & Their Classifying Spaces #

Lie groups are topological groups with a smooth structure, so their classifying spaces (BG) can be constructed to be smooth as well. For classical Lie groups, we have concrete geometric models for (BG):

• (BO(n)) (classifying space for the orthogonal group (O(n))) is the infinite-dimensional Grassmannian of (n)-dimensional real subspaces of (\mathbb{R}^\infty).
• (BU(n)) (classifying space for the unitary group (U(n))) is the infinite-dimensional complex Grassmannian of (n)-dimensional complex subspaces of (\mathbb{C}^\infty).

These Grassmannians have well-understood cohomology rings, computed via Schubert cell decompositions. Characteristic classes (like Stiefel-Whitney classes for (BO(n)), Chern classes for (BU(n))) are exactly the generators of these cohomology rings—so studying (H^*(BG; R)) directly gives you the algebraic structure of all possible characteristic classes.

Vector Bundles & Classifying Spaces #

The link between vector bundles and classifying spaces is straightforward:

• An (n)-dimensional real vector bundle over (X) is equivalent to an (O(n))-principal bundle (by taking the bundle of orthogonal frames for each fiber).
• This means vector bundle isomorphism classes correspond bijectively to homotopy classes of maps (X \to BO(n)).

Characteristic classes tie into this by being universal cohomology classes:

• Stiefel-Whitney classes (w_i \in H^i(BO(n); \mathbb{Z}/2\mathbb{Z})) pull back via the classifying map (f: X \to BO(n)) to give the Stiefel-Whitney classes (f^*w_i \in H^i(X; \mathbb{Z}/2\mathbb{Z})) of the vector bundle.
• Similarly, Chern classes (c_i \in H^{2i}(BU(n); \mathbb{Z})) pull back to Chern classes of complex vector bundles.

By understanding the cohomology ring of (BG), you automatically get all the key properties of characteristic classes—like how they multiply, behave under bundle sums, or restrict to subbundles—because these properties are encoded in the algebraic structure of (H^*(BG)).

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