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Homotopy theory, homological algebra, algebraic treatments of manifolds.

4 votes

what is the image of $\partial( 1_{S^n})$ for the exact sequence for the fibration $X \to E ...

Any fibration $X \to E \to S^n$ up to fiber homotopy equivalence is given by the "clutching construction" applied to a map $f: S^{n-1} \to G(X)$, where $G(X)$ denotes the topological monoid of self ho …
John Klein's user avatar
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3 votes

Examples for open disc bundle which is not vector bundle

Some remarks which amount in some way to an answer. (1) Let $\text{Diff}(D^n)$ be the diffeomorphisms of $D^n$ which restrict to the identity on the boundary. When $n\gg k$ is large, $\pi_k(\text{D …
John Klein's user avatar
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4 votes

Acyclic complexes for extraordinary cohomology theories

If you go to exotic cohomology with twisted coefficients, then the answer is yes. Alternatively, one can state the condition on the level of a universal cover and then the answer is yes. It is certa …
John Klein's user avatar
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2 votes

Fibrations with non-simply connected base and rational homology

I'm wondering the extent to which the assumptions can be tweaked. Let's assume $B$ is connected and with basepoint. Let $F$ be the fiber over the basepoint. However, I won't assume $F$ is homotopy …
John Klein's user avatar
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13 votes
Accepted

Reverse map of a homology equivalence.

As I was writing this answer, Oscar beat me to the punch. I will keep it posted anyway. Let $X^3$ be the Poincare homology sphere. Let $\tilde X \to X$ be the universal cover (note: $\tilde X$ is $S^ …
John Klein's user avatar
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3 votes

Does the classifying space of monoids commute with wedge sum up to weak equivalence?

Charles has given a very good answer to the question. The following is not meant to be an answer, but just a heuristic argument which I cannot make into a proof. There should be an operation, "fr …
John Klein's user avatar
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9 votes

Can both G and BG be finite CW complexes?

The answer is always no unless $G$ is trivial. In fact, I can generalize your statement slightly: I only need to assume that $G$ and $BG$ have merely the homotopy type of finite complexes. I will ou …
John Klein's user avatar
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10 votes

When can you desuspend a homotopy cogroup?

Hopkins' result alluded to above gives a coordinate-free approach a lá Segal. The paper I wrote with Schwänzl and Vogt: Comultiplication and suspension. Topology Appl. 77 (1997), no. 1, 1–18, gives …
John Klein's user avatar
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10 votes

The fundamental group of space which has both an H and a co-H structure

Another argument which does it: A path connected based space $X$ is a co-$H$ space if and only if the evaluation map $\Sigma \Omega X \to X$ admits a section up to homotopy. This will imply that $\p …
John Klein's user avatar
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10 votes
Accepted

Contraction of a family of loops simultaneously

Surely not. Let $S^1 \to LS^2$ be adjoint to the map $c: S^1 \times S^1 \to S^2$ which collapses $S^1\vee S^1$ to a point. The latter has degree one. Let $p\in S^1$ be any point but the basepoint. …
John Klein's user avatar
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14 votes

How to prove the connected sum of two closed aspherical n-manfolds (n >2) is not asperical?

Here's a different way to see it. Let $M$ and $N$ be aspherical of dimension at least 3. Then the wedge $M \vee N$ is aspherical (but not a manifold). Let $M\sharp N$ be the connected sum. Then we get …
John Klein's user avatar
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4 votes
Accepted

Extreme rigidification of homotopy self-equivalences

This is a substantial revision of my original post. It shows that if we replace the "equivalence" Tyler is asking for by a "retract" then the answer is yes. Given a CW space $Y$, we can take $G(Y) = …
5 votes

Homotopy type of the plane minus a sequence with no limit points

Here is a more naive solution, as least if the sequence is countable. Let $\Bbb N \subset \Bbb R^2$ be the embedding defined by the sequence. Then there is an isotopy from this embedding to the stand …
John Klein's user avatar
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10 votes
Accepted

Algebraic K-theory of odd-dimensional spheres

Let $\tilde A(X)$ be the reduced functor, i.e., the homotopy fiber of the map $A(X) \to A(\ast)$. Since $A(*)$ is rationally a product of $K(Q,4j+1)$ for $j \ge 1$, we may as well study $\tilde A(X)$ …
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5 votes

Obstruction Theory for Vector Bundles and Connections

Correction: the definition below is wrong. It isn't true that a 1-flat reduction is the same as a flat reduction. One also needs to require that the map $Z \to BG$ factors through $BG^\delta$, where $ …
John Klein's user avatar
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16 votes

Choice of base point in a Waldhausen category

Although it might seem surprising to some readers, Waldhausen was not going for abstraction merely for the sake of itself in making his definitions: he had concrete applications in mind (most importan …
John Klein's user avatar
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9 votes
Accepted

Homotopy Units in $A_\infty$-spaces

For your first question: If $X$ has the homotopy type of a CW space, then you can replace $X$ by any CW space $Y$ that is homotopy equivalent to it (in the unbased sense). Then $Y$ is also $A_\inft …
John Klein's user avatar
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16 votes

What does actually being a CW-complex provide in algebraic topology?

One should not forget that certain invariants, such as Whitehead torsion, are defined using a choice of CW structure.
John Klein's user avatar
  • 18.5k
8 votes

what does BG classify? i.e. what is a principal fibration?

When $G$ is discrete, another sort of answer is provided by the paper of Michael Weiss: What does the classifying space of a category classify? Homology, Homotopy and Applications 7 (2005), 185–195. …
John Klein's user avatar
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6 votes
Accepted

Is $S^1\vee S^1$ an Eilenberg-Mac Lane Space to a Homotopy Purist?

I think the following can be turned into a proof, but I haven't checked the details. By a result of Milnor, $\Omega (S^1 \vee S^1)$ coincides up to homotopy with $F(S^0 \vee S^0)$, the free group fu …
John Klein's user avatar
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1 vote

Components of a loop space, semidirect products, and multiplicativity

No in general. Yes if and only if the fibration $\tilde X \to X \to BG$ is trivializable, where $X\to BG$ classifies the universal cover $\tilde X$. (Let me assume here that $X$ is connected.) For if …
John Klein's user avatar
  • 18.5k
1 vote

$G$-equivariant intersection theory using differential topology?

You may want to take a look at Klein, J.R., Williams, B. Homotopical intersection theory, II: equivariance. Math. Z. 264(2010),849–880. An arXiv version appears here: https://arxiv.org/abs/0803.0017 I …
John Klein's user avatar
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7 votes

weak equivalence of simplicial sets

The answer is no, and there are plenty of counterexamples. Note that simplicial sets are not relevent here; one can cook up examples with spaces and take their total singular complexes. For example, …
John Klein's user avatar
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10 votes

Homotopy Groups of Connected Sums

Here is something that's valid in the stable range. If $M$ and $N$ are closed $n$-manifolds, there is a cofibration sequence $$ S^{n-1} \to M_0 \vee N_0 \to M\sharp N $$ where $M_0$ denotes the effec …
John Klein's user avatar
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1 vote

Regular homotopy invariance of Wall's self-intersection form.

Even more is true in the context of surgery theory: let $q \ge 3$ with $q$ odd, assume $M$ is a $1$-connected closed smooth manifold of dimension $2q$. Let $$I^{\text{fr}}_q(M)$$ denote the space of i …
John Klein's user avatar
  • 18.5k
1 vote

On a special case of Alexander duality

The proof that $S^n \setminus K$ is path connected follows directly from general position if $K$ is, say, a finite simplicial complex of codimension two (I don't know what "codimension two" means in y …
John Klein's user avatar
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1 vote

Tubular neighborhoods of chains

Here's an approach which might work (I'm not sure about the correctness of this.) 1) Assume $M$ is closed. Choose a triangulation $T$ of $M$. If the support of $c$ is contained inside the $p$-skelet …
John Klein's user avatar
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8 votes

What characteristic class information comes from the 2-torsion of $H^*(BSO(n);Z)$?

See Brown, Edward, The cohomology of $B\text{SO}_n$ and $B\text{O}_n$ with integer coefficients. Proc. Amer. Math. Soc. 85 (1982), no. 2, 283–288 for an answer to your questions. (As for applications: …
John Klein's user avatar
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3 votes

Why does the map $BG\to A(*)$ fail to split?

I just realized how the argument for Question 1 might go (I hope this isn't self-indulgence on my part): The composite $BO \to BG \to A(*)$ factors through $Q(S^0)$. So it suffices to show there is …
John Klein's user avatar
  • 18.5k
2 votes
Accepted

Relating two notions of geometric realization

Regarding What is the relation between $|K|$ and $X$, more specifically between $|\sigma|$ and $X_\sigma$ for each simplex $\sigma \in K$? It seems to me that one can build an intermediate space …
John Klein's user avatar
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