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Section 2 of this paper of Rezk addresses exactly the question of when the localization by S yields a Cartesian model category. For that the relevant property is that that if you take the product of a …

Here is a way to see it which might constitute "just unfolding the definitions" (at least if one were sitting in on the class at Harvard when it was being taught). Set $Z(T) = Y(T^c)$, (compliment tak …

The "pants" bordism in dimension n is a bordism which goes from $S^n \sqcup S^n$ to $S^n$ witnessing the connected sum operation - equivalently by attaching 1-handle to the trivial bordism, equivalent …

$\DeclareMathOperator\Hom{Hom}$It is well-known (see Breen, Mikhailov, Touzé - Derived functors of the divided power functors for example) that for $A$ a free abelian group we have $$ H_i(K(A,1); \mat …

In even dimensions $n=2k$ we can define two smooth manifolds $M$ and $N$ to be stably diffeomorphic if they become diffeomorphic after the connect sum with $r$ many copies of $S^k \times S^k$ for some …

Recall that two 4-manifolds $M$ and $N$ are stably diffeomorphic if there exist $m,n$ such that $$M \#_n (S^2 \times S^2) \cong N \#_n (S^2 \times S^2).$$ That is, they become diffeomorphic after taki …

The completeness condition is not really about making things invertible which weren't already. It is about where the information about invertible morphisms is stored. We can already see this with $(\i …

The Pontryagin-Thom construction allows one to identify the stable homotopy groups of spheres with bordism classes of stably normally framed manifolds. A stable framing of the stable normal bundle ind …

Given a category $\mathcal{C}$ enriched in spaces, we can take the nerve (a simplicial space) and then geometric realization to get a space $B\mathcal{C}$. If we view spaces as $\infty$-groupoids, the …

It is independent of the choice of base point. Let $Map((X,x_0), (G_F,id))$ be the based mapping space (based at $x_0$). Let $Map(X, G_F)$ be the free mapping space. Then we have a split short exact …

This is not true in general, unless you assume the base is sufficiently nice (eg a CW-complex). Here is a counter-example. Let $B = \mathbb{Q}$, the rationals with its topology as a subspace of the …

I looked at Wall's paper Classification problems in differential topology. V On certain 6-manifolds. In theorem 3 of that paper Wall describes some invariants of 6-mainfolds, and the relation between …

I glanced at the paper briefly. Let me try to explain what I understand. Let us suppose that M is compact and without boundary. Let $f: M \to \mathbb{R}$ be a Morse function. Let us further suppose fo …

As requested I am writing this as an answer. No there are spaces with vanished homology which are not homotopy equivalent to finite CW-complexes. For example if $G$ is an acyclic group, then the cl …

Here is an argument, which is basically Denis Nardin's comment. To have a model independent proof you need model independent definitions of the hocolim and of the localization. You can define them …