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Results tagged with at.algebraic-topology
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user 8032
Homotopy theory, homological algebra, algebraic treatments of manifolds.
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' …
- 18.5k
34
votes
Accepted
Can anyone explain to me what is an assembly map?
If you are given a homotopy functor $L$ from spaces to spectra, the assembly map is a natural map of spectra
$$
H_\bullet(X;L) \to L(X) ,
$$
where the domain is a homology theory. This homology theory …
- 18.5k
28
votes
0
answers
1k
views
On the (derived) dual to the James construction.
Background
If $X$ is a based space then the James construction on $X$ is the space $J(X)$ given by
$$
X \quad \cup \quad X^{\times 2} \quad \cup \quad X^{\times 3} \quad \cup \quad \cdots
$$
in …
26
votes
Accepted
Groupoid actions on spaces
Perhaps the most natural example is given by universal covers?
Let $X$ be a "nice" space. For a point $x\in X$ let $\tilde X_x$ be the universal covering of
$X$ taken at $x$ (the fiber at $y \in X$ …
- 18.5k
22
votes
A map inducing isomorphisms on homology but not on homotopy
How about a knot complement? It's known that the inclusion of a meridian $S^1$ (a small circle which links the knot exactly once) into the knot
complement $S^3 \setminus S^1$ is a homology isomorphis …
- 18.5k
19
votes
Is any CW complex with only finitely many nonzero homology groups homotopic to a finite dime...
In the simply connected case, the answer is yes.
In the general case, the theory was worked out in complete detail by Wall in the paper:
Wall, C. T. C. Finiteness conditions for CW-complexes. Ann. of …
- 18.5k
19
votes
Fibrations and Cofibrations of spectra are "the same"
The following might help answer the last part of your post:
In the late 1960s, Tudor Ganea developed technology that studies the difference between the homotopy fibers and cofibers of a map. For exam …
- 18.5k
18
votes
What is the intuition behind the Freudenthal suspension theorem?
The Freudenthal theorem is really a special case of the phenomenon called "homotopy excision" aka the Blakers-Massey triad theorem. The idea is that one has an inclusion
$$
(C_-X,X) \to (\Sigma X,C_+X …
- 18.5k
18
votes
Accepted
Realizing cohomology classes by submanifolds
Your question is just a reformulation of what Thom did, so the answer is always yes.
Since the Stokes map from de~Rham cohomology to singular cohomology (with real coefficients) is an isomorphism, yo …
- 18.5k
18
votes
Lambda-operations on stable homotopy groups of spheres
The operation which sends a finite set $S$ to its set of $k$-element subsets, $\binom{S}{k}$, gives rise to the $k$-th stable Hopf invariant. There is additional structure in this: the set $\binom{S}{ …
- 18.5k
18
votes
1
answer
567
views
Local homology of a space of unitary matrices
Let $U(n)$ denote the unitary group (this is a manifold of dimension $n^2$). Let
$$
{\cal D} \subset U(n)
$$
denote the subspace of those matrices having
a non-trivial $(+1)$-eigenspace.
Backgroun …
- 18.5k
18
votes
2
answers
2k
views
Geometric interpretation of the Pontryagin square
The Pontryagin square (at the prime 2) is a certain cohomology operation
$$
\mathfrak P_2: H^q(X;\Bbb Z_2) \to H^{2q}(X;\Bbb Z_4)
$$
which has the property that its reduction mod 2 coincides with $x\m …
- 18.5k
17
votes
Accepted
The classifying space of a gauge group
Proof of (1):
(a). Suppose $X$ and $Y$ are $G$-spaces, the action of $G$ on $X$ is free, and $X\to X/G$ is
a principal bundle, then the space of $G$-equivariant maps
$$
F(X,Y)^G
$$
is the same th …
- 18.5k
17
votes
3
answers
2k
views
Transversality in the proof of the Blakers-Massey Theorem. Is it necessary?
Assume one is given a commutative square of spaces
$A \quad \to \quad C$
$
\downarrow \qquad \qquad \downarrow$
$B\quad \to \quad X$
which is a pushout and in which each map is a cofibratio …
- 18.5k
16
votes
Choice of base point in a Waldhausen category
Although it might seem surprising to some readers, Waldhausen was not going for abstraction merely for the sake of itself in making his definitions: he had concrete applications in mind (most importan …
- 18.5k
16
votes
What does actually being a CW-complex provide in algebraic topology?
One should not forget that certain invariants, such as Whitehead torsion, are defined
using a choice of CW structure.
- 18.5k
16
votes
Is an A-infinity thing the same the same as strict thing viewed through a homotopy equivalence?
The answer to your question about the loop space is a conditional "yes." The conditions are:
i) X should have the homotopy type of a CW complex, and
ii) When we say topological group, we mean with …
- 18.5k
16
votes
1
answer
2k
views
On the wikipedia entry for Borel-Moore homology
The wikipedia page on Borel-Moore homology claims to give several definitions of it,
all of which are supposed to coincide for those spaces $X$ which are homotopy equivalent to a finite CW complex and …
- 18.5k
16
votes
1
answer
544
views
Whitehead products and Framed Manifolds
The attaching map for the top cell of the torus $S^n \times S^n$ is a map
$$
[x,y]: S^{2n-1} \to S^n \vee S^n
$$
where the notation is such that
$x,y : S^n \to S^n \vee S^n$ are the two inclusions––– …
- 18.5k
16
votes
4
answers
1k
views
Multiplicativity of the Euler characteristic for fibrations
For a Serre fibration
$$
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 …
- 18.5k
15
votes
Computational complexity of computing homotopy groups of spheres
Here's a very useless algorithm due to Kan: Let $G(S^n)$ be the simplicial group that is
Kan loop group of the $n$-sphere. In each simplicial degree, it is a free group.
(This simplicial group has the …
- 18.5k
15
votes
2
answers
1k
views
"Strøm-type" model structure on chain complexes?
Background
The Quillen model structure on spaces has weak equivalences given by the weak homotopy equivalences and the fibrations are the Serre fibrations. The cofibrations are characterized by t …
- 18.5k
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 …
- 18.5k
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 …
- 18.5k
14
votes
Accepted
Characteristic classes for block bundles
I don't know where the results are written down in one place (perhaps in the book of Madsen and Milgram?), but see the the end of this post for a list of references.
In any case, here is a proof of …
- 18.5k
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^ …
- 18.5k
13
votes
do spectra have diagonal maps?
The existence of an $E_\infty$-diagonal is an obstruction for equipping a spectrum $E$ with the structure of a suspension spectrum. Conversely, in
Klein, J.R.: Moduli of suspension spectra. Trans. A …
- 18.5k
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 …
- 18.5k
12
votes
1
answer
209
views
A variant of $\ell^2$-cochains
Suppose $X$ is an infinite countable CW complex which satisfies the following property: for all $k$-cells $e$, the number of $(k+1)$-cells incident to $e$ is at most $c_k$, where the latter is some n …
- 18.5k
12
votes
Accepted
Definition of Pontrjagin Classes
The odd Chern classes of the complexified bundle are of order 2 and are determined by the Stiefel-Whitney classes of the original real bundle $\xi$ by the formula
$$
c_{2k+1}(\xi\otimes \Bbb C) = \bet …
- 18.5k
12
votes
Accepted
Is the J homomorphism compatible with the EHP sequence?
Added 9/7/16:
I just got access to the paper:
James, I. M. On the iterated suspension. Quart. J. Math., Oxford Ser. (2) 5, (1954). 1–10
which is an explicit reference to Greg's questions on the leve …
- 18.5k
12
votes
The space of framed functions
Eliashberg once told me that the framed function theorem should be a consequence of his work on wrinkled maps.
Igusa and I gave a fairly direct proof that the space of framed functions on the circle …
- 18.5k
12
votes
Accepted
"C choose k" where C is topological space
I presume you are asking is whether one can make sense of $\binom{X}{k}$ as a space in such a way that it relates to the formula you are asking about.
The answer is yes.
First let $l = 1$. For a sp …
- 18.5k
12
votes
1
answer
844
views
Equivariant homotopy theory: some history questions
I have sometimes wondered about the following:
(1) Who was the first articulate that in dealing with $G$-equivariant cohomology theories ($G$ a finite group or a compact Lie group), it is best to wor …
- 18.5k
12
votes
0
answers
394
views
Hilton-Eckmann dual of the Steenrod Algebra
In essence my question can be stated as follows: fill in the analogy
$$
\text{cup product} \qquad\qquad \leftrightarrow \qquad \text{Samelson product}
$$
$$
\updownarrow \qquad\qquad \qq …
- 18.5k
12
votes
1
answer
895
views
On the stable splitting of loops on a suspension
Let $X$ be a connected, based CW complex. Then the James splitting
of $\Sigma\Omega\Sigma X$ gives, in particular, a weak equivalence of spectra
$$
\Sigma^{\infty} \Omega\Sigma X_+ \quad \simeq \quad …
- 18.5k
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
$$
K( …
- 18.5k
12
votes
Why should I prefer bundles to (surjective) submersions?
You write:
So, I'm wondering for some applications where I really need to use a bundle --- where some important fact is not true for general submersions (or, surjective submersions with connected fi …
- 18.5k
11
votes
Homotopy classification of selfmaps of product of spheres?
No chance. For example, take the self maps of $S^3 \times S^3$. Then based maps gives
$$
\text{maps}_\ast(S^3\times S^3,S^3 \times S^3) = \text{maps}_\ast(S^3\times S^3,S^3) \times \text{maps}_\ast(S …
- 18.5k
11
votes
Does the Borel functor take equivariant fibrations to fibrations?
I believe what you are asking is true whenever $X\to B$, considered as an unequivariant map,
is a Serre fibration.
First some definitions:
Call a map of $G$-spaces $E \to B$ a $G$-Serre fibration …
- 18.5k
11
votes
topological type of smooth manifolds with prescribed homotopy type and pontryagin class
In the $1$-connected case, one may argue as follows:
Let $X$ be a closed $1$-connected smooth $n$-manifold, $n \ge 5$. The theory of the Spivak fibration shows that any homotopy equivalence $f: M^n …
- 18.5k
11
votes
Intuition behind Alexander duality
Yet another way to get it. Alexander duality is closely related to Poincare duality: suppose we can write
$$
S^n = K \cup_A C
$$
where $K$ and $C$ are codimension zero compact manifolds with common bo …
- 18.5k
11
votes
1
answer
583
views
co-$A_\infty$ spaces
A co-$A_n$ space is a based space $Y$ equipped with a co-action by the Stasheff associahedron operad $K_\bullet$. This means that $Y$ is comes with certain maps $c_n: Y \times K_n \to Y^{\vee n}$, $n …
- 18.5k
11
votes
Accepted
Linking topological spheres
Let $C = S^3 \setminus A$. Alexander duality says that
$$
H_1(C) \cong H^1(A) \cong \Bbb Z\, .
$$
Let $\alpha: S^1 \to C$ be any map representing a generator of $H_1(C)$ (every first homology class …
- 18.5k
11
votes
2
answers
665
views
Reference request: Goodwillie tower of the identity
The Taylor (Goodwillie) tower of the identity functor on based spaces has as its $j$-th layer the infinite loop space-valued functor
$$
X\mapsto \Omega^\infty (W_j \wedge_{h\Sigma_j} X^{[j]})
$$
in wh …
- 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^ …
- 18.5k
11
votes
Natural transformations as categorical homotopies
Concerning
Have anyone ever introduced natural transformation in this "homotopical" way rather then the classical one in any reference like a textbook or some lecture notes?
Yes, Quillen introdu …
- 18.5k
11
votes
Occurrences of (co)homology in other disciplines and/or nature
A classical and elegant application is to the solution of Kirchhoff's theorem on electrical cricuits. See:
Nerode, A.; Shank, H.: An algebraic proof of Kirchhoff's network theorem.
Amer. Math. Month …
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 $\tilde A(X)$ …
- 18.5k
10
votes
1
answer
588
views
Acyclic aspherical spaces with acyclic fundamental groups
A space $X$ (by which I mean a CW complex) is acyclic if its reduced singular homology $\tilde H_\ast(X;\Bbb Z)$ is trivial in all degrees.
A discrete group $\pi$ is said to be acyclic if its classi …
- 18.5k