Search Results

It is false for $d\geq 2$. For an example with $d=2$, start with a circle, and attach two 2-discs to it via (for example) a map of degree 4 and a map of degree 5. If you like to think in terms of gr …

The given conditions imply that the map $\psi:L\rightarrow X$ induces an isomorphism of fundamental groups, and so it lifts to a map $\tilde\psi: \widetilde L\rightarrow \widetilde X$. The conditions …

Every finite simplicial complex has the homotopy type of a triangulable manifold with boundary, and triangulated manifolds are pure. So the condition `pure' doesn't restrict the possible homotopy typ …

I think that you are missing the definition of 'orientable along $A$'. I haven't got that book of Bredon to hand, but presumably 'orientable along $A$' means that if you move a local orientation of …

Here is a partial answer. There are finite acyclic 2-complexes that are not contractible. Any such 2-complex can be embedded in $\mathbb{R}^5$, and a regular neighbourhood of such a 2-complex will b …

I needed a slightly more complicated example than this in a paper of mine, so I included a proof there. For any non-trivial finite group $Q$ I give a 3-dimensional rationally acyclic complex with a …

It is not true in general. Let $n=3$, and take a piece of a brick wall (thickness one brick) that is homeomorphic to a 3-disc. For example, just take a largish rectangular wall. There are bricks of …

The $E_1$ page does not tell you what the higher differentials will be, and you will have to know at least something about the integral cohomology. Consider the case when $X$ is a Moore space $M(1,\m …

It is true that a group $G$ is $FP_2$ if and only if $G$ acts freely cellularly on a connected CW-complex with trivial first homology group. I see from the comments that you are worried about finite …

This does not answer Greg's question, but it is related. You can realize any $\mathbb{Z}G$-module you that like as $H_1$ of a based 2-complex, or as $H_2$ of a 3-complex if you insist that the comple …

Let $G=(\mathbb{Q},+)$. Then ${\rm cd}(G)=2$ and ${\rm cd}(G\times G)=3$.

Take a surface (without boundary but not compact) of infinite genus, and tesselate it by polygons so that there are no non-trivial combinatorial symmetries of the surface. The universal covering of …

If all vertex links in a finite simplicial complex $K$ are homology $n-1$-spheres (i.e., homeomorphic to $n-1$-manifolds with the same homology as an $n-1$-sphere), then the simplicial complex $K$ is …

Yes, there is such a decomposition. The spheres on which $\mathbb{Z}/p\mathbb{Z}$ acts freely are necessarily odd dimensional for $p>2$. View each $S^{2m-1}$ as the unit sphere in $\mathbb{C}^m$, wi …

This answers a slightly different question. If you do not insist on `simply connected' there are examples to show that this does not happen in general. Fix $k\geq 2$. For any $k$, there is a group …