25
votes

### When does homology represent an embedded sphere?

(I'm assuming that the dimension $\dim(M)$ should have been different than the dimension of the homology class--otherwise this can only happen if $M$ is the disjoint union of $S^i$ with some other ...

24
votes

Accepted

### Homotopy equivalent Postnikov sections but not homotopy equivalent

This is a pretty well-known phenomenon, linked with phantom maps.
One of the first existence results was Brayton Gray's paper
Spaces of the same $n$-type, for all $n$, Topology
5 (1966) 241--243
...

17
votes

Accepted

### Classification of closed 3-manifolds with finite first homology group?

The answer is no by Yves' comments. Let me add that there are plenty of explicit constructions of closed hyperbolic 3--manifolds with finite homology, and this is a generic phenomenon (for example ...

17
votes

Accepted

### Is there a theorem showing that de Rham homology is isomorphic to singular homology?

I guess that by de Rham homology you mean the homology groups $H_{k, \, \mathrm{dR}}(X)$ constructed on a closed manifold $X$ by using the complex of currents.
In that case, [1, Theorem 2 page 582] ...

16
votes

### Spectra with "finite" homology and homotopy

Here are two ways of thinking about it. The first comes from the way one proves the final statement you cited: if $X$ has finitely many nonzero homotopy groups which are all finitely generated, then ...

15
votes

### Does homology have a coproduct?

$\def\ZZ{\mathbb{Z}}\def\RR{\mathbb{R}}\def\PP{\mathbb{P}}$I noticed these excellent answers were missing an explicit example of a space $X$ and a class $\eta$ in $H_{\ast}(X, \ZZ)$ such that the ...

15
votes

Accepted

### Smallest volume representatives of homology

Finding volume-minimizing representatives of homology classes is one of the major applications of geometric measure theory.
It's a theorem of Federer and Fleming that any nontrivial integral $k$-...

15
votes

Accepted

### When homology isomorphism implies homotopy isomorphism

Here's a counterexample.
Set $X'=S^1\vee S^2$.
Consider the following map $F':S^2\vee S^2\vee S^2\rightarrow X'$: It maps the first $S^2$ summand to the $S^2$ summand of $X'$ via a map that ...

14
votes

Accepted

### Geometric intuition behind this chain homotopy

When thinking about chain homotopies in a setting involving simplices it can be helpful to consider the product $\Delta^p\times I$ where $\Delta^p$ is a $p$-simplex and $I=[0,1]$. The formula $h\...

14
votes

Accepted

### loop space of a finite CW-complex

This is true for finite $\pi_1$ and false for infinite $\pi_1$: Let $\widetilde{X}$ denote the universal cover of $X$, then $\Omega\widetilde{X}$ is the unit connected component of $\Omega X$, and $\...

14
votes

Accepted

### What is the rank of the period lattice of modular forms?

For Q1: I'm assuming your $f$ is a normalized Hecke newform of level $N$ with coefficients in $\mathbf{Z}$. Then the choice of $f$ determines a splitting of the homology as
$$H_1(X_0(N), \{cusps\}, \...

13
votes

### Integral homology classes that can be represented by immersed submanifolds but not embedded submanifolds

This is a great question, and I don't have an answer but this is too long for a comment.
Working mod $2$, a codimension $k$ homology class $z\in H_{m-k}(M;\mathbb{Z}/2)$ is realizable by an embedding ...

12
votes

Accepted

### Impossibility of realizing codimension 1 homology classes by embedded non-orientable hypersurfaces

If $[\Sigma]\in H_{n}(M,\mathbb Z)$ is primitive, its image $[\Sigma] \in H_{n}(M,\mathbb Z/2\mathbb Z)$ can always be represented by a non-orientable manifold! Since it is primitive, we may suppose ...

11
votes

Accepted

### Understanding Homology Operations and how to compute them

I will work stably: everything in sight will be a spectrum.
It is well known and classical that cohomology operations correspond to map of spectra: that is if $E,F$ are two spectra, a natural ...

11
votes

### Homology $H_*(TOP, \mathbb{Z}_2)$ of the stable homeomorphism space

You should certainly start by looking at the book "The classifying spaces for surgery and cobordism of manifolds" by Madsen and Milgram. There is a PDF copy at http://www.maths.ed.ac.uk/~aar/papers/...

11
votes

Accepted

### Continuous maps $f:S^n \to \mathbb{C}P^m$ with $f(x)\perp f(-x) $

Such a map $S^n\to \mathbb CP^m$ exists if and only if either $n<2m$ or $n=2m=2$.
To see this, first note that such a map is the same as a $\mathbb Z/2$-equivariant map from $S^n$ to a certain ...

11
votes

### When does homology represent an embedded sphere?

As mentioned in Tyler Lawson's answer, the homology class $w \in H_i(M)$ in question needs to be in the image of the Hurewicz map so that there is at least a continuous map $f : S^i \to M$ with $f_*[S^...

11
votes

Accepted

### What is the homotopy type of the poset of nontrivial decompositions of $\mathbf{R}^n$?

Let me write $V$ for a finite-dimensional vector space over some field (the field will not play a role), and $\mathsf{P}(V)$ for the poset described in the question, which I consider as a category. ...

10
votes

Accepted

### Two surfaces in a 4-manifold whose algebraic intersection number is zero

Yes, this can be done by tubing one surface along the other.
Suppose that you have two intersection points $p_+, p_- \in \Sigma_1 \cap \Sigma_2$ of opposite signs. Suppose also that $\Sigma_1$ and $\...

10
votes

### Does the cohomology Bockstein homomorphism map to the homology Bockstein homomorphism under Poincarè duality?

I'm not sure I quite understand your formulation of the question (what, for example, is a section $s\colon A_3 \to A_1$?) but the following is probably what you're looking for. Given a closed oriented ...

10
votes

### Integral homology classes that can be represented by immersed submanifolds but not embedded submanifolds

I don't know if you'll find this a satisfying example, but what about $2\in \mathbb{Z} \cong H_1(S^1)$? This can be represented by the immersion that wraps the circle twice around itself, but it can't ...

9
votes

Accepted

### Homology of the product of spaces with integer coefficients and the Massey products

Here's a very general form. Suppose that we have six chain complexes $A_0, A_1, A_2, A_{01}, A_{12}, A_{012}$, with bilinear "multiplication" pairings of chain complexes:
$$
\begin{align*}
A_0 \otimes ...

9
votes

Accepted

### Software for computing Thurston's unit ball

"Better late than never." Stephan Tillmann and William Worden have produced the software package tnorm. This can be found here:
https://pypi.org/project/tnorm/
The software should be able ...

9
votes

### Relative homology of free loop space with respect to constant loops

Let me tackle the case that $Q$ is $1$-connected, but not $2$-connected. Because $Q$ is $1$-connected we have that $\Lambda_0Q=\Lambda Q$, as all loops are contractible.
Two sequences are relevant ...

9
votes

Accepted

### Explicit $BP_*BP$-comodule structure on $BP_*\mathbb{C}P^n$ and $BP_*\mathbb{C}P^{\infty}$

A concise formula
$$
\mu(\beta) = \beta(c(t^F))
$$
for the $BP_* BP$-coaction $\mu$ on $BP_* CP^\infty$ is given in the "Note added in proof" on page 279 of
...

9
votes

Accepted

### Integral homology classes of which no multiples admit embedded representatives with trivial normal bundle

With the trivial normal bundle condition, it's fairly easy to produce non-realizable examples using Théorème II.2 of Thom's paper. Namely, a class $z\in H_l(M^n;\mathbb{Z})$ is realizable by an ...

9
votes

### Cohomology of finite symmetric products of manifolds

$H^*(SP^n(M);\mathbb{Q}) \cong (H^*((M)^n;\mathbb{Q}))^{\Sigma_n} \cong (H^*(M;\mathbb{Q})^{\otimes n})^{\Sigma_n}$
Note that the tensor product here (via the Kunneth formula) is a graded tensor ...

8
votes

### Why torsion is important in (co)homology ?

The following is an example of how torsion is used in the theory of 3-manifolds.
It is a theorem of Thurston that there are infinitely many non isometric compact hyperbolic 3-dimensional manifolds of ...

8
votes

### Homology spectral sequence for function space

An alternative spectral sequence for $H_∗(map_∗(X,Y);k)$ can be constructed using the approach in this paper of mine (apols for self-promotion). Actually, this spectral sequence is discussed already ...

8
votes

Accepted

### On spaces with finite homological dimension

You can take $X=BG$ where $G$ is a torsion-free, acyclic group of infinite cohomological dimension. Acyclic means $H_k(BG;\mathbb{Z})=0$ for $k>0$, and infinite cohomological dimension implies ...

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