All Questions
9,056 questions
12
votes
2
answers
2k
views
Why the choice of the simplex for defining homology?
This question arose out of my attempts to understand another question. The most popular construction for the chain complex for defining singular homology uses the $n$-simplex.
But it is also possible ...
CommunityBot
- 1
1
vote
1
answer
2k
views
Thom isomorphism
Let$p:E\rightarrow B$ be an n dimensional vector bundle, R be a commutative ring. Assume that B is simplyconnected or char R=2. Then there is an element $U\in H^n(M(p);R)$ such that we have dual ...
- 21
1
vote
1
answer
170
views
Does there exists a (possibly homological) characterization of the Jordan curve property in all dimensions?
More precisely, let $M$ be a subspace $\mathbb R^n$ with the following properties:
$M$ is a topological manifold of dimension $n-1$.
M is compact.
Does there exist a homological characterization of ...
- 60.5k
4
votes
2
answers
810
views
Cohomology of complex projective spaces with coefficientes in a complex-orientable cohomology theory
Hello everyone,
I'm having problems understanding a basic fact about complex-orientable cohomology theories:
Let $E^{\ast}$ be a multiplicative cohomology theory and $x\in E^2({\mathbb C}\text{P}^{\...
- 27.2k
15
votes
3
answers
2k
views
Survey articles on homotopy groups of spheres
Are there general surveys or introductions to the homotopy groups of spheres? I'm interested especially in connections to low-dimensional geometry and topology.
- 8,467
4
votes
1
answer
363
views
Homotopy colimits of cyclic spaces
Let $\Lambda$ denote Connes's cyclic category. It is an extension of the simplex category $\Delta$ (of nonempty finite linearly ordered sets) obtained by adding an automorphism of order $n+1$ to the ...
- 24.1k
14
votes
2
answers
2k
views
Ordinary cohomology of stacks
Let $\mathbf{X}$ be a stack over $Top$ (a lax sheaf of groupoids, or some such thing). If it admits a surjective representable map $F \to \mathbf{X}$ then one can form the iterated fibre product to ...
- 6,229
3
votes
1
answer
2k
views
Does there exist a classification of covering spaces in algebraic geometry?
This is a question based on the heuristics that most things in algebraic/differential topology has an analogue in algebraic geometry.
The fundamental group classifies the covering spaces of a (...
- 4,015
9
votes
2
answers
2k
views
Functoriality of fundamental group via deck transformations
Problem
I'm trying to understand this with a view towards the etale fundamental group where we can't talk about loops. What I'm missing is how the fundamental group functor should work on morphisms, ...
CommunityBot
- 1
14
votes
2
answers
1k
views
Dyer-Lashof based spectral sequence for homotopy classes of maps between infinite loop spaces (spectra).
The homology of an infinite loop space, which represents a spectrum, is an algebra over the Dyer-Lashof algebra (see for example Cohen-Lada-May's Springer volume, or for part of the story the more ...
- 498
3
votes
1
answer
292
views
Cartesian-closed category of spaces with the Whitehead property?
I'm not sure if this is standard, but we'll call the property that every weak homotopy equivalence is an honest homotopy equivalence the Whitehead property (from Whitehead's theorem for CW complexes). ...
- 15.5k
6
votes
1
answer
2k
views
When does a "representable functor" into a category other than Set preserve limits?
This might be a dumb question. If $C$ is an ordinary category, then for any $c \in C$ the covariant representable functor $\text{Hom}(c, -) : C \to \text{Set}$ preserves limits. However, it can ...
CommunityBot
- 1
5
votes
1
answer
230
views
Nonuniqueness of maps of representing spaces
In Rudyak's On Thom Spectra, Orientability, and Cobordism, two variants of Brown's representability theorem are presented: given a natural transformation $f^*: E^* \to F^*$ of cohomology theories, ...
- 26.8k
7
votes
4
answers
6k
views
Simplicial complexes vs. geometric realization of abstract simplicial complexes
A finite abstract simplicial complex is a pair $D=(S,D)$ where $S$ is a finite set and $D$ is a non-empty subset of the power set of $S$ closed under the subset operation, e.g. $(\{a,b,c\},\{\emptyset,...
- 21.3k
5
votes
2
answers
1k
views
Why is the intersection of complex submanifolds always positive.?
Hi, everyone:
I was finally able to show that all complex manifolds are orientable, by
generalizing to many variables the fact that , for a single complex variable,
the Jacobian matrix is ...
- 14.4k
4
votes
1
answer
934
views
Pontrjagin square (and possible typo in Mosher & Tangora?)
There's an exercise at the end of Ch. 2 of Mosher & Tangora's "Cohomology Operations and Applications in Homotopy Theory", which says:
Suppose the cocycle $u\in C^{2p}(X;Z)$ satisfies $\delta u=...
- 52.6k
8
votes
2
answers
658
views
Cofinal inclusions of Waldhausen categories
Let $\mathcal{C}$ be a Waldhausen category. Suppose that $\mathcal{B}$ is a subcategory of $\mathcal{C}$, and that $\mathcal{B}$ is closed under extensions. If $\mathcal{B}$ is strictly cofinal in $\...
- 6,229
4
votes
2
answers
937
views
CW structure on spaces obtained by attaching cells wildly
Is there necessarily a CW structure on a space build out of cells without demanding them to be attached in "right" order?
More precisely, let $X$ be a topological space such that the map $\emptyset\...
- 8,803
5
votes
2
answers
878
views
What is an example of a non-regular, totally path-disconnected Hausdorff space?
I need this for a counterexample: the multiplication in the fundamental group $\pi_1(\Sigma X_+)$, when it is equipped with the topology inherited from $\Omega \Sigma X_+$, fails to be continuous for ...
- 7,193
16
votes
2
answers
2k
views
What is known about K-theory and K-homology groups of (free) loop spaces?
Calculating the homology of the loop space and the free loop space is reasonably doable. There exists the Serre spectral sequence linking the homology of the loop space and the homology of the free ...
- 12.1k
0
votes
1
answer
470
views
Natural embedding GL_n(C) -> C^{n^2} \ {0} induces zero on cohomology
The question looks like an exercise in elementary algebraic topology, but I didn't manage to solve it. I am considering this question because it is a toy example in a problem I'm thinking about.
Let'...
CommunityBot
- 1
18
votes
2
answers
790
views
The kernel of the map from the handlebody group to Outer automorphisms of a free group
Let $K$ be a compact oriented 3-dimensional handlebody of genus $g$. The group $H_g$ of isotopy classes of diffeomorphisms of $K$ is called the handlebody group. (It embeds as a subgroup of the ...
- 28.2k
15
votes
3
answers
14k
views
How to demonstrate $SO(3)$ is not simply connected?
A quote from Wikipedia's article on the Rotation group:
Consider the solid ball in $\mathbb{R}^3$ of
radius $\pi$ [...].
Given the above, for every point in
this ball there is a rotation, ...
- 4,394
7
votes
1
answer
458
views
Why do Delta-sets not allow quotients?
A $\Delta$-set is a contravariant functor from the category $\Delta'$ of order-preserving injections to the category of sets (this is essentially what Allen Hatcher calls a $\Delta$-complex).
A main ...
CommunityBot
- 1
11
votes
1
answer
671
views
Is it always possible to compute the Betti numbers of a nice space with a well-chosen Lefschetz zeta function?
Let $X$ be a smooth projective variety. If I've understood correctly, the Weil conjectures imply that it is possible to compute the Betti numbers of $X(\mathbb{C})$ by computing the local zeta ...
CommunityBot
- 1
10
votes
1
answer
635
views
Self-homomorphisms of surface groups
Let $X$ be a closed, orientable surface of genus at least 2, and let $\phi: \pi_1(X) \to \pi_1(X)$ be a surjective homomorphism. Is $\phi$ necessarily injective?
- 44.8k
4
votes
1
answer
1k
views
Twisting an L-infinity-morphism with "non-associated" Maurer-Cartan elements
Background
Suppose we are given $L_\infty$-algebras $(g,Q)$ and $(g',Q')$ and an $L_\infty$-morphism $F$ from $(g,Q)$ to $(g',Q')$. Furthermore, we have a Maurer-Cartan element $\pi$ of $(g,Q)$.
One ...
- 2,869
18
votes
1
answer
1k
views
Fundamental groups of the spaces of rational functions
Here is a question which I asked myself (and couldn't answer) while reading "The topology of spaces of rational functions" by G. Segal.
Let $X$ be a smooth complete complex curve (=a compact Riemann ...
9
votes
0
answers
700
views
Is a functor which is a sheaf for open covers and finite closed covers automatically a sheaf for covers by simplices?
Suppose $F:Top^{op}\rightarrow Set$ is a functor which is a sheaf for open coverings as well as for finite closed coverings, i.e. such that, apart from the usual sheaf property we also have that for ...
- 12.3k
14
votes
4
answers
4k
views
Oriention-Reversing Diffeomorphisms of a Manifold
I am trying to figure out when a closed, oriented manifold admits an orientation reversing diffeomorphism. My naive argument that the orientation cover should allow you to switch orientations is ...
- 967
8
votes
0
answers
438
views
Differential K-theory computation
I am trying to read about K-theory and differential K-theory. I understand that the K-theory of spheres can be computed explicitly (by getting to the stable range and using Bott periodicity and so on)....
- 3,383
4
votes
2
answers
598
views
Can H-space multiplication always be straightened so that mult.-by-id. is the identity on the nose?
In JP May's Concise Course in Algebraic Topology, on page 143 he says that the left- and right-multiplication-by-identity maps $\lambda:X\rightarrow X$ and $\rho:X\rightarrow X$ specify a map $X\vee X\...
- 20k
9
votes
2
answers
763
views
Knot complement diffeomorphism groups and embedding spaces
I'm interested in the following collection of questions: Let $S^n_k = \sqcup_k S^n$ be a disjoint union of $k$ distinct $n$-dimensional spheres. Write $Emb(S_k^n, S^{n+2})$ for the space of ...
CommunityBot
- 1
5
votes
1
answer
539
views
Relation of Lie Groups and Cohmology Theories via Formal Group Laws
There is a standard process (for example explained here) to obtain a formal group law form a complex oriented cohomology theory.
For a Lie group G one can choose coordinates at the unit and expand ...
- 6,162
21
votes
0
answers
1k
views
What is the current knowledge of equivariant cohomology operations?
In Caruso's paper, "Operations in equivariant $Z/p$-cohomology," http://www.ams.org/mathscinet-getitem?mr=1684248, he shows that the integer-graded stable cohomology operations in $RO(\mathbb{Z}/p)$-...
- 865
7
votes
1
answer
826
views
Weight filtration for smooth analytic manifolds
In his ICM 2002 talk (Topology of singular algebraic varieties, available also on arXiv) B. Totaro says on p. 3 (of the arXiv version): "Using the method of Guillen and Navarro Aznar I was able to ...
- 35.2k
8
votes
2
answers
1k
views
Poincaré quasi-isomorphism
Suppose we have a simplicial combinatorial manifold (just a triangulated manifold) and its Poincaré dual cell complex.
Corresponding homology simplicial and homology cell complexes are quasi-...
- 1,482
10
votes
1
answer
847
views
Applications of Faber's conjecture
Faber's perfect pairing conjecture states that the tautological ring $R^*$ of the moduli space $\mathcal{M}_g$ of curves of genus $g$ behaves as if it were the rational cohomology of a closed, ...
- 13.1k
12
votes
1
answer
1k
views
classification of smooth involutions of torus
Let $\mathbb{Z}_2=\{1,g\},T^2=\{(e^{i\theta_1},e^{i\theta_2})\}$ and place $T^2$ in $\mathbb{R}^3$ as the locus of the rotation of $2\pi$ rads of the circle$\{(y,z)|(y-2)^2+z^2=1\}$ around $z$ axis.
...
- 28.2k
40
votes
4
answers
3k
views
Chain homotopy: Why du+ud and not du+vd?
When one wants to prove that a morphism $f_*$ between two chain complexes $\left(C_*\right)$ and $\left(D_*\right)$ is zero in homology, one of the standard approaches is to look for a chain homotopy, ...
- 33.8k
5
votes
3
answers
3k
views
Topological description of Manifold with boundary
Is there any 'general' topological invariant to tell the difference between $M$ and $N$, where $M$ has homotopy type of a closed manifold and $N$ has homotopy type of a manifold with boundary.
I ...
- 22.1k
22
votes
1
answer
2k
views
Formal-group interpretation for Lin's theorem?
Background
For compact Lie groups, Atiyah and Segal proved a strong relationship between Borel-equivariant K-theory, defined in terms of the K-theory of $X \times_G EG$, and the equivariant K-theory ...
- 27.2k
2
votes
2
answers
601
views
Must a Strong deformation retractible 3-manifold be homeomorphic to $\mathbb{R}^3$?
Assume $M$ is an open 3-manifold which can be deformation retracted to a point. Is it necessarily homeomorphic to $\mathbb R^3$?
(I know Whitehead had an example which is contractible and not ...
- 44.7k
24
votes
1
answer
5k
views
Do "surjective" degree zero maps exist?
Is there a map $f\colon X \to Y$ of closed, connected, smooth and orientable $n$-dimensional manifolds such that the degree of $f$ is 0 but $f$ is not homotopic to a non-surjective map?
Added: The ...
- 3,376
6
votes
2
answers
985
views
Lifting a homeomorphism, always possible?
Let $h:M\to M$ be a homeomorphism of a compact manifold. Let $p:\tilde M\to M$ be a covering. 1) Is it always possible to lift $h$ to $H:\tilde M\to \tilde M$ so that everything fits into the ...
- 44.7k
11
votes
2
answers
777
views
Presentation of the monoid of surfaces
In the following every surface is assumed to be connected.
I've read that the commutative monoid of homeomorphism classes of closed surfaces is generated by $P$ (projective plane) and $T$ (torus) ...
CommunityBot
- 1
10
votes
1
answer
1k
views
Verb form of 'homotopy'? 'Homotope'?
Is there a transitive verb, in common use, which means 'deform via a homotopy'? I used to think 'homotope' was the answer, but it produces surprisingly few relevant matches on Google, so now I have ...
- 12.8k
2
votes
1
answer
533
views
Given a ramified cover of a Riemann surface, is there a good choice of basis for H_1 of the source?
Suppose we are given a map $f: X \to Y$ between two Riemann Surfaces, with branch points $p_1,p_2,\dots,p_n$ and known multiplicities at these points. Assuming we have a basis of $H_1(Y, \mathbb{Z})$,...
- 19.2k
3
votes
2
answers
344
views
Principle when limits level by level don't commute with simplicial structure
Are there general principles when a simplicial object is a (co)limit of other simplicial objects level by level, but is not a (co)limit when considering the entire simplicial structure?
Objects can ...
- 1,975
39
votes
1
answer
5k
views
Flatness in Algebraic Geometry vs. Fibration in Topology
I am currently trying to get my head around flatness in algebraic geometry. In particular, I'm trying to relate the notion of flatness in algebraic geometry to the notion of fibration in algebraic ...
- 57.6k