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Repeating the first part of Oscar's answer and elaborating on comments by Chris and Panagiotis, here is a down-to-earth argument in all cases: The cases $n=1,3,7$ are fine, since the stable stems are …

Here's an argument that works in the non simply-connected case and avoids the universal cover and spectral sequences assuming the existence of rationalizations. Assume that $X$ has rational cohomolog …

To avoid leaving this question open: Assuming we work in the category of compactly generated spaces, geometric realization commutes with pullbacks.(It's crucial that we use the compactly generated pr …

Stiefel-Whitney numbers detect (unoriented) bordism classes and together with Pontryagin numbers, they determine oriented bordism classes. Ranicki's total surgery obstruction of a finite $n$-dimensio …

As John Klein remarked, the answer to this question will depend on the classifying space functor $B$ one uses. Let me present one case for which the question can be answered positive which is basical …

The mapping class group of a smooth manifold $M$ is the group of all its self diffeomorphisms up to isotopy, i.e. $\pi_0(\operatorname{Diff}(M))\cong \pi_1(B\operatorname{Diff}(M))$. A large portion …

Every connected Liegroup, which has a semisimple Liealgebra with a definite Killing form is compact. The Liealgebra of a compact Liegroup is always the direct sum of an semisimple and abelian Liealgeb …

Just for completeness, here's another argument without spectral sequences via rational homotopy theory. Recall a theorem of Hopf, which states that the rational cohomology of a path-connected H-space …

The classification problem of smooth oriented closed $(n-1)$-connected $2n$-manifolds for $n\ge3$ splits into three parts. Classify smooth almost closed compact oriented $(n-1)$-connected $2n$-manifo …

Since my comment answered Emanuele Dotto's answer, I post it as an answer: Stefan Schwede discusses equivariant bordism in his book project about global homotopy theory in detail.

If everything takes place in the category of compactly generated spaces, it holds $$\pi_0(BC)=\pi_0(obC)/\tilde{},$$ where two path components in the object space get identified, if there are objects …

Let me assume that M is at least 5-dimensional. Sullivan's proof only uses surgery theory and properties of O(n) that also hold for Top(n), so the answer to your first question is yes. Regarding your …

For many values of $n$, the answer to both questions is no. Since the fundamental groups of $BX$ and $B\mathrm{Diff}(S^n)$ are finite for $n\ge5$ (This uses that $\pi_0\mathrm{Diff}_\partial(D^n)$ is …

Appendix B of Randal-Williams' "Cohomology of automorphism groups of free groups with twisted coefficients" gives a stable description of the graded $\mathbb{Q}[\Sigma_q]$-module $H^*(\Gamma_g;H^{\oti …

I'm unsure whether this question is appropriate for mathoverflow, so feel free to criticize. All manifolds are closed, smooth and have dimensions $n\ge 5$. The Atiyah-Shapiro-Bott-Orientation gives …