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Homotopy theory, homological algebra, algebraic treatments of manifolds.

3 votes
Accepted

Explaining some detail in Wall's paper of CW-complexes

As to (1): If we choose a basepoint in $K$, then $\phi$ can be viewed as a map of based spaces. Let $F$ be the homotopy fiber of $\phi$. Then there is a well-defined action $\Omega K \times F \to F$ w …
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6 votes
Accepted

Is $\Sigma^\infty_+ O(n)^\vee$, the Spanier-Whitehead dual of the orthogonal group, an $A_\i...

The question as stated probably requires clarification. If $X$ is a space, then the S-dual $D_+(X)$ (i.e., functions from $X_+$ to the sphere) is always an $E_\infty$-ring spectrum. In particular, it …
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8 votes
1 answer
202 views

Finite domination and Poincaré duality spaces

Here are some definitions: A space is homotopy finite if it is homotopy equivalent to a finite CW complex. A space finitely dominated if it is a retract of a homotopy finite space. A space $X$ is a Po …
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8 votes
1 answer
753 views

On the Euler characteristic of a Poincaré duality space

Background. Suppose that $M$ is an oriented, connected, closed manifold of dimension $d$ with fundamental class $\mu \in H_d(M;\Bbb Z)$. Let $\Delta : M \to M \times M$ be the diagonal map. Then the p …
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13 votes
2 answers
489 views

"Burnside ring" of the natural numbers and algebraic K-theory

The construction of the Burnside ring $A(G)$ of a group $G$ (usually, but not always, finite) is given by taking the Grothendieck group of the commutative semi-ring of isomorphism classes of finite $G …
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8 votes
1 answer
310 views

Finite domination and compact ENRs

Edit: In the comments, Tyrone points out that West's positive answer to Borsuk's conjecture implies that every compact ENR is homotopy equivalent to a finite CW complex. It follows that the only finit …
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2 votes

Multiplicativity of the Euler characteristic for fibrations

Here is an argument that the Euler characteristic is multiplicative for fibrations $$ F\to E \to B $$ where $F$ and $B$ are finitely dominated and $B$ is connected. Without loss in generality, we may …
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7 votes

Multiplicativity of the Euler characteristic for fibrations

Note Added March 1, 2022: I now think there is a gap in deducing multiplicativity of the Euler characteristic from the Pedersen-Taylor result on the finiteness obstruction. I think the argument I giv …
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16 votes
4 answers
1k views

Multiplicativity of the Euler characteristic for fibrations

For a Serre fibration $$ F\to E \to B , $$ with $F,E,B$ having the homotopy type of finite complexes, it is known that the Euler characteristic is multiplicative: $$ \chi(E) = \chi(F)\chi(B) . $$ Howe …
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61 votes
2 answers
3k views

Thomason's "open letter" to the mathematical community

In 1989, Bob Thomason left his CNRS position in Orsay and moved to Paris VII. It was during this period that he composed his "Open Letter" to the mathematical community. The letter explained Thomason' …
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8 votes

How can I construct a closed manifold from a finite CW complex?

More generally, suppose $n \le m$ are non-negative integers, $X$ is a CW complex of dimension $\le n$, $M$ is a non-empty, closed $m$-manifold, and $X$ and $M$ have the same homotopy type. It is well …
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2 votes
Accepted

Retractions, homology and multiplication

The role of $C$ in the problem is irrelevant. Let $r : C \to B$ be a retraction, and set $$m := r\circ \mu : B \times B \to B . $$ Set $\ast := b$ and think of it as the basepoint of $B$. Then the r …
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9 votes
Accepted

Atiyah duality without reference to an embedding

Here is another short construction which is much simpler and just takes a few lines. Let $M$ be a closed $n$-manifold. Consider the diagonal $M \to M \times M$. It is an embedding. Take its Pontryagi …
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7 votes

Atiyah duality without reference to an embedding

Assume $M$ is a closed, smooth manifold of dimension $n$. Let $\tau^+$ be the fiberwise one point compactification of its tangent bundle. This is a fiberwise $n$-spherical fibration equipped with a p …
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7 votes
1 answer
564 views

Filtered homotopy colimits and singular homology

Suppose I have a functor $$ X_\bullet: I \to \text{Spaces} $$ where $I$ is a small filtered category. It seems to be a "folk theorem" that the homomorphism $$ \underset{\alpha\in I}{\text{colim }} H_ …
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10 votes
Accepted

Whitehead product and a homotopy group of a wedge sum

Here are some details which are related to Tyler's comment. I recommend looking at the paper "Induced Fibrations and Cofibrations" by Tudor Ganea (1967). For connected based spaces $X$ and $Y$, there …
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7 votes
0 answers
132 views

Weak homotopy type of the Cech Nerve

Let ${\cal U} = \{U_i\}_{i\in J}$ be an open cover of a topological space $X$, where the indexing set $J$ is assumed to be well-ordered. Then the Cech nerve is the "simplicial space without degeneraci …
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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 …
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12 votes
1 answer
335 views

Rational homotopy invariance of algebraic $K$-theory

Suppose that $R\to S$ is a 1-connected morphism of connective structured ring spectra that induces an isomorphism on rational homotopy groups. Is the induced map of (Waldhausen) K-theory spectra $$ K( …
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7 votes

Why does this construction give a (homotopy-invariant) suspension (resp. homotopy cofiber) i...

An argument showing that the two models of suspension are equivalent will probably be based on something like the following: Assertion: Suppose we are given a commutative diagram of the form $\requir …
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3 votes
Accepted

A generalization of integral Poincaré duality

Prior to Dwyer-Greenlees-Iyengar, Dwyer and myself (independently) defined Gorenstein conditions for group rings over the sphere $S[G]$, i.e., the suspension spectrum of a topological group. The def …
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3 votes

Is it true, the space of embeddings segments is homotopy equivalent to the subspace of all l...

Here are some remarks which may be relevant. First of all it seems to me that the correct topology to use is the Whitney $C^\infty$-topology on the embedding space. Let $M$ be an closed manifold. T …
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10 votes

When is a homotopy pushout contractible?

Let me add some additional remarks on the enumeration question: How many such spaces $A$ are there sitting over $B\times C$ such that the homotopy pushout $$ B \leftarrow A \to C $$ is contractible …
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5 votes
Accepted

pullback and fiber sequence

Yes. Here are some details. The space $P$ sits in homotopy pullback diagram $\require{AMScd}$ $$ \begin{CD} P @>>> D \\ @VVV@VVV \\ A\times C @>>> D\times D \end{CD} $$ where the the right vertica …
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2 votes

K-theory of free $G$-sets and the classifying space, and generalization

I believe you meant to write $Q(BG_+)$ in the first paragraph of your post, where $Q = \Omega^\infty\Sigma^\infty$. The this result is really a folk theorem and is sometimes called the "Barratt–Priddy …
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3 votes
Accepted

CW complexes obtained by attaching cells not with increasing dimension

There is a name for the kind of space you are describing: a cell complex. A CW complex is a cell complex which has cell attachments in the increasing order of dimension. The main advantage of having …
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4 votes

moving from sphere spectrum to finite spectrum

For convenience I will set $Z = Z^k$, where $Z^k$ is as in your notation. Case 1: $Y$ is the suspension spectrum of a based finite complex $U$ having dimension $n$. Then $\pi^Y_\ast(Z)$ is given by …
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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. …
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10 votes

Why is $\mathbb{S}^1$ a cogroup object in $\mathbf{Top.}$?

The circle is not a cogroup object in the category of based spaces. If $X$ is a cogroup in based spaces, then the composition $$ X \to X \vee X \to X\times X $$ would necessarily coincide with the di …
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14 votes

Unstable manifolds of a Morse function give a CW complex

(1). Some experts tell me that Laudenbach's paper is incomplete and contains gaps. I will retract this for now. I do recall being told this, but I am not aware at this point in time where the gaps …
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5 votes

Does anyone know a basepoint-free construction of universal covers?

Here is another attempt at pinning down the meaning of "canonical" in reference to Tom's answer. Let $X$ be a nice space (connected, locally path-connected and semi-locally simply connected). Let …
John Klein's user avatar
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6 votes
0 answers
123 views

On the weak homotopy type of a differentiable (Chen) space

Suppose that $M$ is a differentiable space in the sense of Chen (cf. https://ncatlab.org/nlab/show/Chen+space ). Assume that $M$ also has the structure of a topological space and that the two struct …
John Klein's user avatar
  • 18.5k
3 votes
Accepted

Wall self-intersection invariant for odd-dimensional manifolds?

I doubt that what you are proposing as the receptacle for the obstruction is the correct abelian group. For one thing, you are not taking into account the involution on the canonical double cover of t …
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7 votes
Accepted

Turning injection of homotopy groups to an isomorphism

Your question is equivalent to the following: Given a cellular inclusion $i : X\to Y$, when is there a retraction $r:Y \to X$? (Being a retraction means that $r\circ i: X\to X$ is the identity.) T …
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7 votes
Accepted

What is the homotopy fiber of $X \to X_{hG}$, where this is a pointed homotopy orbit?

Here is a special case which gives a partial answer: (i). Suppose $G$ acts in a homotopically trivial way on $X$. This means that there is a trivial $G$-space $Y$ and a pair of $G$-equivariant maps …
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  • 18.5k
7 votes

Homology of the universal cover

If we replace the field $k$ with the ring of integers $\Bbb Z$, then no. There are non-trivial high dimensional knots $K: S^n \to S^{n+2}$, whose complements $X = S^{n+2}-K(S^n)$ have $\pi_1(X) \con …
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18 votes
Accepted

Realizing cohomology classes by submanifolds

Your question is just a reformulation of what Thom did, so the answer is always yes. Since the Stokes map from de~Rham cohomology to singular cohomology (with real coefficients) is an isomorphism, yo …
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  • 18.5k
6 votes

Notion of linking between two general $p$ and $q$ manifolds embedded in a higher dimensional...

There are various approaches to this. One approach, developed by Bruce Williams and me, uses homotopy theory. See: Homotopical intersection theory, I. Geometry & Topology 11, (2007) 939–977 arXiv: m …
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2 votes
Accepted

Relation between transport functor of a fibration and a Hurewicz connection on it

Let $p: E\to B$ be a map. Define $\Lambda(p) = E \times_B B^I$; this is the space of pairs $(x,\gamma)$ consisting of a point $x\in E$ and a path $\gamma: [0,1] \to B$ such that $\gamma(0) = p(x)$. Th …
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  • 18.5k
6 votes
Accepted

$X \rtimes Y \simeq X \vee (X \wedge Y)$ for $X$ a co-H-Space

The proof is not hard, but more tedious than I would have thought. There is a canonical identification $$ X\rtimes Y = X\wedge(Y_+) $$ where $Y_+$ is the effect of adding a disjoint base point to $ …
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  • 18.5k
6 votes
1 answer
256 views

An operad-like structure, is there a name for it?

Here is an example which I'd like to have a name for. Let $P$ be a compact smooth manifold of dimension $p$, possibly with non-empty boundary. Define $E(k,P)$ to be the space of smooth (codimension …
John Klein's user avatar
  • 18.5k
3 votes

Homotopy type of smooth manifolds with boundary

I know of two proofs in the compact case. Let $M$ be a compact smooth $m$-manifold with boundary $\partial M$. 1) Morse theory (Sketch). For this I think we need $m \ge 4$. There is a Morse functio …
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6 votes

Can one disjoin any submanifold in $\mathbb R^n$ from itself by a $C^{\infty}$-small isotopy?

There are lots of counter-examples. Here's one: The fibration $\text{SO}(3) \to \text{SO}(4) \to S^3$ splits, since it is the principal bundle of the tangent bundle of $S^3$, and the latter is paral …
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  • 18.5k
11 votes
Accepted

Connectivity of suspension-loop adjunction

If the spectrum $X$ is $r$-connected, then the map $\Sigma^\infty\Omega^\infty X \to X$ is $(2r+2)$-connected. Here's a sketch: apply the functor $\Omega^\infty$ to get the map of spaces $$ Q(\Omega^ …
John Klein's user avatar
  • 18.5k
6 votes

Homology spectral sequence for function space

Let $X$ be a simplicial set and $Y$ a space (or simplicial set) . Then $F_\ast(X,Y)$ is a cosimplicial space and we can consider its homology spectral sequence. Bousfield gave conditions for when this …
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  • 18.5k
4 votes

A map of spaces implementing the Pontryagin Thom collapse map? (collapse maps in families)

The situation is somewhat easier to describe if one replaces the embeddings of the closed codimension $(m-n)$manifold $M$ with the embeddings of the total space of a disk bundle of a rank $(m-n)$-vect …
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  • 18.5k
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 $ …
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6 votes
Accepted

Cup products in the Mayer-Vietoris sequence

Here are some comments. The map $\delta^\ast$ is a composition of the form $\require{AMScd}$ \begin{CD} H^{\ast-1}(U\cap V) @> \sigma >\cong > H^{\ast}(\Sigma (U\cap V)) \to H^\ast X \end{CD} wher …
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10 votes
Accepted

The space of contractible loops of a finite dimensional $K(\pi,1)$

The statement is true for a $K(\pi,1)$ but not true for other $X$. Finite dimensionality is not relevant. Here's a sketch: let $X = K(\pi,1)$ and I will assume $X$ has the homotopy type of a CW com …
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  • 18.5k
5 votes

What do Atiyah and Segal mean by $K_G^*(X)$?

Although not given in the Atiyah-Segal paper, I suspect the definition is as follows: The functor $K_G$ is defined on pairs of $G$-spaces $(X,A)$, since it is a cohomology theory (I guess it's the Gr …
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