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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 ...
Tyler Lawson's user avatar
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24 votes
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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 ...
Jeff Strom's user avatar
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17 votes
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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 ...
Jean Raimbault's user avatar
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] ...
Francesco Polizzi's user avatar
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 ...
skd's user avatar
  • 5,590
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 ...
David E Speyer's user avatar
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 $k$-...
j.c.'s user avatar
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15 votes
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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 ...
John Greenwood's user avatar
14 votes
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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\...
Allen Hatcher's user avatar
14 votes
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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 $\...
Achim Krause's user avatar
  • 9,114
14 votes
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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\}, \...
David Loeffler's user avatar
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 ...
Mark Grant's user avatar
  • 35.4k
12 votes
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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 ...
Bruno Martelli's user avatar
11 votes
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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 ...
Denis Nardin's user avatar
  • 16.3k
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/...
Neil Strickland's user avatar
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 ...
Tom Goodwillie's user avatar
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^...
Michael Albanese's user avatar
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. ...
Oscar Randal-Williams's user avatar
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 $\...
Marco Golla's user avatar
  • 10.5k
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 ...
Dave Benson's user avatar
  • 13.6k
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 ...
Greg Friedman's user avatar
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 ...
Tyler Lawson's user avatar
  • 51.9k
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 ...
Sam Nead's user avatar
  • 26.7k
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 ...
Thomas Rot's user avatar
  • 7,403
9 votes
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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 ...
John Rognes's user avatar
  • 9,163
9 votes
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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 ...
Mark Grant's user avatar
  • 35.4k
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 ...
Ian Agol's user avatar
  • 67.7k
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 ...
Gregory Arone's user avatar
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 ...
Mark Grant's user avatar
  • 35.4k

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