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The original space was not really a simplicial set.
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Sebastian Goette
Sebastian Goette
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There isFor a counterexample if $X$ is allowed to be, we glue two-dimensional Moore spaces for the geometric realisation of a simplicial set with nondegenerate simplices of dimensionsgroups $\le d$ only,$\mathbb Z/2$ and $d=2$ is enough$\mathbb Z/3$ along a common $S^1$. But I don't see how this would work forAs a traditional simplicialCW complex described, $X$ can be realised by gluing two 2-disks into a subset ofclosed loop $\gamma$, such the power set of some finite setboundaries of verticesthe disks wind around $\gamma$ two and three times, respectively.

TakeTo get a 1simplicial complex, start with a triangle, then glue in a simplicial 6-simplex $\sigma$gon and collapse its boundary to a pointsimplicial 9-gon as above (both need a sufficiently fine simplicial structure for this to work). Then glue in two Removing a 2-simplices. Forsimplex produces a space that is homotopy equivalent the first, identify each faceCW complex above with $\sigma$ preserving the natural orientations. Forone of the second, treat two disks removed, that is, to one of the three faces as above and collapse the third to a pointtwo Moore spaces.

The fundamental group has a generator $[\sigma]$$[\gamma]$ and two relations $3[\sigma]=0$$2[\gamma]=0$ and $2[\sigma]=0$$3[\gamma]=0$, so $X$ is simply connected. If you take away one of the two 2-simplicesdisks, you loose one relation, so you loose simple connectivity.

The associated cellular chain complex looks like $$\mathbb Z\stackrel{0}\longleftarrow\mathbb Z\stackrel{(3,2)}\longleftarrow\mathbb Z^2\;,$$ and one checks that $H_2(X)\cong\mathbb Z$, so $X$ is not contractible.

Edit. The example above can be turned into a simplicial complex in the traditional sense. Simply replace the loop $\sigma$ by a triangle, then glue in a simplicial 6-gon and a simplicial 9-gon as described above (both need a sufficiently fine simplicial structure for this to work). Removing a two simplex produces a space that is homotopy equivalent to one with one of the two disks removed.

There is a counterexample if $X$ is allowed to be the geometric realisation of a simplicial set with nondegenerate simplices of dimensions $\le d$ only, and $d=2$ is enough. But I don't see how this would work for a traditional simplicial complex described by a subset of the power set of some finite set of vertices.

Take a 1-simplex $\sigma$ and collapse its boundary to a point. Then glue in two 2-simplices. For the first, identify each face with $\sigma$ preserving the natural orientations. For the second, treat two of the three faces as above and collapse the third to a point.

The fundamental group has a generator $[\sigma]$ and two relations $3[\sigma]=0$ and $2[\sigma]=0$, so $X$ is simply connected. If you take away one of the two 2-simplices, you loose one relation, so you loose simple connectivity.

The associated chain complex looks like $$\mathbb Z\stackrel{0}\longleftarrow\mathbb Z\stackrel{(3,2)}\longleftarrow\mathbb Z^2\;,$$ and one checks that $H_2(X)\cong\mathbb Z$, so $X$ is not contractible.

Edit. The example above can be turned into a simplicial complex in the traditional sense. Simply replace the loop $\sigma$ by a triangle, then glue in a simplicial 6-gon and a simplicial 9-gon as described above (both need a sufficiently fine simplicial structure for this to work). Removing a two simplex produces a space that is homotopy equivalent to one with one of the two disks removed.

For a counterexample, we glue two-dimensional Moore spaces for the groups $\mathbb Z/2$ and $\mathbb Z/3$ along a common $S^1$. As a CW complex, $X$ can be realised by gluing two 2-disks into a closed loop $\gamma$, such the boundaries of the disks wind around $\gamma$ two and three times, respectively.

To get a simplicial complex, start with a triangle, then glue in a simplicial 6-gon and a simplicial 9-gon as above (both need a sufficiently fine simplicial structure for this to work). Removing a 2-simplex produces a space that is homotopy equivalent the CW complex above with one of the two disks removed, that is, to one of the two Moore spaces.

The fundamental group has a generator $[\gamma]$ and two relations $2[\gamma]=0$ and $3[\gamma]=0$, so $X$ is simply connected. If you take away one of the two disks, you loose one relation, so you loose simple connectivity.

The associated cellular chain complex looks like $$\mathbb Z\stackrel{0}\longleftarrow\mathbb Z\stackrel{(3,2)}\longleftarrow\mathbb Z^2\;,$$ and one checks that $H_2(X)\cong\mathbb Z$, so $X$ is not contractible.

Also works as simplicial complex.
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Sebastian Goette
Sebastian Goette
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This seems to depend on the notion of simplicial complex. There is a counterexample if $X$ is allowed to be the geometric realisation of a simplicial set with nondegenerate simplices of dimensions $\le d$ only, and $d=2$ is enough. But I don't see how this would work for a traditional simplicial complex described by a subset of the power set of some finite set of vertices.

Take a 1-simplex $\sigma$ and collapse its boundary to a point. Then glue in two 2-simplices. For the first, identify each face with $\sigma$ preserving the natural orientations. For the second, treat two of the three faces as above and collapse the third to a point.

The fundamental group has a generator $[\sigma]$ and two relations $3[\sigma]=0$ and $2[\sigma]=0$, so $X$ is simply connected. If you take away one of the two 2-simplices, you loose one relation, so you loose simple connectivity.

The associated chain complex looks like $$\mathbb Z\stackrel{0}\longleftarrow\mathbb Z\stackrel{(3,2)}\longleftarrow\mathbb Z^2\;,$$ and one checks that $H_2(X)\cong\mathbb Z$, so $X$ is not contractible.

Edit. The example above can be turned into a simplicial complex in the traditional sense. Simply replace the loop $\sigma$ by a triangle, then glue in a simplicial 6-gon and a simplicial 9-gon as described above (both need a sufficiently fine simplicial structure for this to work). Removing a two simplex produces a space that is homotopy equivalent to one with one of the two disks removed.

This seems to depend on the notion of simplicial complex. There is a counterexample if $X$ is allowed to be the geometric realisation of a simplicial set with nondegenerate simplices of dimensions $\le d$ only, and $d=2$ is enough. But I don't see how this would work for a traditional simplicial complex described by a subset of the power set of some finite set of vertices.

Take a 1-simplex $\sigma$ and collapse its boundary to a point. Then glue in two 2-simplices. For the first, identify each face with $\sigma$ preserving the natural orientations. For the second, treat two of the three faces as above and collapse the third to a point.

The fundamental group has a generator $[\sigma]$ and two relations $3[\sigma]=0$ and $2[\sigma]=0$, so $X$ is simply connected. If you take away one of the two 2-simplices, you loose one relation, so you loose simple connectivity.

The associated chain complex looks like $$\mathbb Z\stackrel{0}\longleftarrow\mathbb Z\stackrel{(3,2)}\longleftarrow\mathbb Z^2\;,$$ and one checks that $H_2(X)\cong\mathbb Z$, so $X$ is not contractible.

There is a counterexample if $X$ is allowed to be the geometric realisation of a simplicial set with nondegenerate simplices of dimensions $\le d$ only, and $d=2$ is enough. But I don't see how this would work for a traditional simplicial complex described by a subset of the power set of some finite set of vertices.

Take a 1-simplex $\sigma$ and collapse its boundary to a point. Then glue in two 2-simplices. For the first, identify each face with $\sigma$ preserving the natural orientations. For the second, treat two of the three faces as above and collapse the third to a point.

The fundamental group has a generator $[\sigma]$ and two relations $3[\sigma]=0$ and $2[\sigma]=0$, so $X$ is simply connected. If you take away one of the two 2-simplices, you loose one relation, so you loose simple connectivity.

The associated chain complex looks like $$\mathbb Z\stackrel{0}\longleftarrow\mathbb Z\stackrel{(3,2)}\longleftarrow\mathbb Z^2\;,$$ and one checks that $H_2(X)\cong\mathbb Z$, so $X$ is not contractible.

Edit. The example above can be turned into a simplicial complex in the traditional sense. Simply replace the loop $\sigma$ by a triangle, then glue in a simplicial 6-gon and a simplicial 9-gon as described above (both need a sufficiently fine simplicial structure for this to work). Removing a two simplex produces a space that is homotopy equivalent to one with one of the two disks removed.

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Sebastian Goette
Sebastian Goette
  • 6.8k
  • 2
  • 36
  • 62

This seems to depend on the notion of simplicial complex. There is a counterexample if $X$ is allowed to be the geometric realisation of a simplicial set with nondegenerate simplices of dimensions $\le d$ only, and $d=2$ is enough. But I don't see how this would work for a traditional simplicial complex described by a subset of the power set of some finite set of vertices.

Take a 1-simplex $\sigma$ and collapse its boundary to a point. Then glue in two 2-simplices. For the first, identify each face with $\sigma$ preserving the natural orientations. For the second, treat two of the three faces as above and collapse the third to a point.

The fundamental group has a generator $[\sigma]$ and two relations $3[\sigma]=0$ and $2[\sigma]=0$, so $X$ is simply connected. If you take away one of the two 2-simplices, you loose one relation, so you loose simple connectivity.

The associated chain complex looks like $$\mathbb Z\stackrel{0}\longleftarrow\mathbb Z\stackrel{(3,2)}\longleftarrow\mathbb Z^2\;,$$ and one checks that $H_2(X)\cong\mathbb Z$, so $X$ is not contractible.