Simplicial

8 Some Classical Homotopy Theory

8.1 The Hurewicz Map

Given a ring \(R\), the \(R\)-Hurewicz map for a pointed simplicial set \(X\) is the map \(\pi _n(X,*) \to \pi _n(R[X],*) \to \pi _n(R[X]/R[*]),0) = \tilde {H}_n(X;R)\). For example, for the \(n\)-sphere \(\Delta ^n/\partial \Delta ^n\), the map given by the \(n\)-cell gets sent to the generator \(1 \in \pi _n(K(R,n))\).

To analyse it the Hurewicz map, we will need the cellular formulation of the Serre spectral sequence. To get this, given a fibration \(p:E \to B\), filter \(B\) by skeleta. \(p^{-1}\sk _nB\) gives a filtration of \(R[E]\). We assume again that \(\pi _1\) acts trivially on homology. The Serre spectral sequence will be the associated spectral sequence of this filtration. For every boundary of a nondegenerate cell \(\partial \Delta ^n \to B\), the Serre spectral sequence \(H_*(\partial \Delta ^n;H_*F) \to H_*(p^{-1}(\partial \Delta ^n))\) degenerates, because the inclusions \(\Delta ^{\{0\}} \to \partial \Delta ^n \to \Delta ^n\) splits off the fibre. Thus the \(E_1\)-term of the spectral sequence is \(\tilde {H}^*(\bigvee p^{-1}\Delta /p^{-1}\partial \Delta ^n)\), which is the sum over the nondegenerate \(n\)-cells of \(H^*(F)[n]\). The \(d_1\)-differential is the alternating sum of the boundary maps, giving the \(E_2\)-term as \(H^*(B;H^*(F))\).

The transgression in the Serre spectral sequence can be identified with the map

(-tikz- diagram)

defined on the subset of elements where it makes sense.

Lemma 8.1. If \(Y\) is \(n\)-connected for \(n \geq 0\) then \(\tilde {H}_i(Y) = 0\) for \(i \leq n\).

Proof. We can replace \(Y\) by a minimal Kan complex. It follows that \(\tilde {H}_i(Y)=0\) since there are no nondegenerate cells of dimension \(\leq n\) other than the basepoint, so the normalized complex of \(R[Y]\) vanishes in those degrees. □

Lemma 8.2. Suppose that \(X \to \Omega Y\) is a map of pointed simplicial sets, where \(Y\) is \(n\)-connected for \(n\geq 1\). Then for \(i\leq 2n\) there is a commutative diagram

where \(\tilde {f}\) is the adjoint of \(f\).

Proof. \(S^1\) can be taken to be the simplicial circle \(\Delta ^1/\partial \Delta ^1\). The top isomorphism is via the connecting homomorphism using the fact that \(X\wedge S^1 = CX/X\), where \(CX\) is the cone on \(X\), given by \(X\wedge \Delta ^1\).

To get the lower isomorphism, we use the fibration \(\Omega Y \to PY \to Y\) that, and observe that the transgression has to be an isomorphism in degrees \(\leq 2n\) by Lemma 8.1.

To see that the diagram is commutative, we apply homology to the commutative diagram

to get

The bottom composite is the transgression, and the top composite is the suspension isomorphism. □

Taking \(f\) to be the identity map \(\Omega Y \to \Omega Y\) we obtain:

Corollary 8.3. Let \(Y\) be \(n\)-connected for \(n\geq 1\) and pointed. Then the transgression of the path space fibration for \(i\leq 2n\) is given by

\[ \tilde {H}_i(Y)\xrightarrow {\ee _*^{-1}} \tilde {H}_i((\Omega Y)\wedge S^1)\xrightarrow {\partial } \tilde {H}_{i-1}(\Omega Y)\]

Proposition 8.4. Suppose \(Y\) is a pointed and \(n\)-connected for \(n\geq 1\). Then for \(i\leq 2n\), there is a commutative diagram

Proof. This follows by applying the Hurewicz map to the path space fibration, and using Corollary 8.3 and the commutative diagram

□

Now we will give an explicit description of the Postnikov tower. Consider the functor \(Z\mapsto Z(r)\) that identifies simplices by the equivalence relation that checks whether they agree on the \(r\)-skeleton. There is clearly a natural transformation \(Z \to Z(r) \to Z(r-1)\), giving the Postnikov tower. \(Z(r)\) is called the \(r^{th}\) Postnikov section. Postnikov sections and the Postnikov tower also refer to any maps equivalent to these. This particular model is sometimes called the Moore-Postnikov tower.

Lemma 8.5. The Postnikov tower is a tower of Kan complexes and Kan fibrations. If \(X\) is a Kan complex, then so is \(X(r)\), and \(\pi _i(X) = \pi _i(X(r))\) for \(i \leq r\) and \(\pi _i(X(r)) = 0\) at every basepoint.

Proof. By Lemma 3.21 it suffices to show the statement about homotopy groups and that \(X \to X(r)\) is a minimal Kan fibration. To see this is a Kan fibration, there is clearly a lift to every horn of an \(n\)-simplex for \(n \leq r+1\). for \(n>r+1\) any \(n\)-simplex in \(X(r)\) will restrict to the horn in the right way because of the definition of \(X(r)\), so the lifting property follows from surjectivity of \(X \to X(r)\).

To see the claim about \(\pi _*\), the vanishing result follows because \(\pi _*\) is given by maps from \(\Delta ^n/\partial \Delta ^n\), which have to be trivial for \(n>r\) since the boundary is that of the completely degenerate simplex. It is easy to see that \(\pi _i(X) = \pi _i(X(r))\) for \(i <r\). For \(i = r\), we observe that the same equivalence relation on maps from \(\Delta ^r/\partial \Delta ^r\) have been imposed since simplices have only been identified relative to the boundary. □

We can also use the notation \(X_{\leq r}\) to denote \(X(r)\) and \(X_{\geq r+1}\) to denote the fibre of \(X \to X(r)\).

Lemma 8.6. Let \(X\) be connected. Then the Hurewicz map identifies \(H_1(X;\ZZ )\) with the abelianization of \(\pi _1(X)\).

Proof. We can choose a model for \(X\) having one vertex. Then the fundamental group is generated by \(1\)-simplices modulo the relation \([\sigma ] = [\sigma '][\sigma '']\) when there is a \(2\)-simplex forcing that relation. \(H_1\) has the same generators but with the abelianized relations. □

Lemma 8.7. The simplicial circle \(S^1\) is a \(K(\ZZ ,1)\).

Proof. By Proposition 6.1, its \(\pi _1\) is \(\ZZ \). It suffices to show \(\pi _i(S^1) = 0\) for \(i>1\). To see this, consider the complex \(S^1_{\geq 2}\) that consist of an integer’s worth of \(\Delta ^1\) such that the end of the \(i^{th}\) on is identified with the beginning of the \({i+1}^{th}\) one. The map \(S^1_{\geq 2} \to S^1\) is a Kan fibration, and it is easy to see that \(S^1_{\geq 2}\) has the homotopy type of a point, since the inclusion of any point is anodyne. The fibres are discrete, giving the desired vanishing. □

Theorem 8.8 (Hurewicz). Suppose that \(X\) is \(n\)-connected. Then the Hurewicz map \(\pi _iX \to \tilde {H}_i(X;\ZZ )\) is an isomorphism for \(i \leq n+1\) and a surjection for \(i=n+2\).

Proof. Consider the fibre sequence \(X_{\geq n+2} \to X \to X_{\leq n+1}\). By the naturality of the Hurewicz map and the Serre spectral sequence, we reduce to the case of \(X_{\leq n+1}\). By Proposition 8.4 the isomorphism in dimension \(n+1\) reduces to the case \(n=0\), where it follows from Lemma 8.6.

So assume \(X\) has homotopy groups only in dimension \(n+1\). The map \(X \to \ZZ [X] \to \ZZ [X]_{\leq n+1}\) (where the truncation is taken in simplicial abelian groups) is an equivalence, so \(X\) is WLOG \(K(\pi _{n+1}(X),n+1)\), so it suffices to show \(\tilde {H}_{n+2}(K(A,n+1))\) vanishes for any abelian group \(A\). \(A\) is a filtered colimit of finitely generated abelian groups which \(\tilde {H}_{n+2}\) commutes with, so it suffices to show it for \(\ZZ \) and \(\ZZ /p^i\). For \(\ZZ /p^i\), we can reduce to \(\ZZ /p\) by using the extensions \(\ZZ /p \to \ZZ /p^i \to \ZZ /p^{i-1}\).

Next, we can use Proposition 8.4 to reduce to the case \(n=2\). Using the path fibre sequence \(K(\ZZ ,1) \to *\to K(\ZZ ,2)\) and the Serre spectral sequence, we get the result for \(\ZZ \) using Lemma 8.7. We can use the fiber sequence \(K(\ZZ ,2) \xrightarrow {p} K(\ZZ ,2) \to K(\ZZ /p,2)\) and examine the spectral sequence to get the result for \(K(\ZZ /p,2)\). □

Theorem 8.9 (Freudenthal). Suppose \(X\) is an \(n\)-connected space with \(n\geq 0\). Then the map \(X\to \Omega (X\wedge S^1)\) is \(2n\)-connected.

Proof. For \(i = 0\), we can observe that the map on \(\pi _1\) factors as \(\pi _1(X) \to H_1(X) = H_1(\Omega X\wedge S^1) = \pi _1(\Omega X\wedge S^1)\), so the result follows by Lemma 8.6.

For \(i>0\), we examine the triangle

and use Corollary 8.3 to learn that it induces an isomorphism in \(\tilde {H}_i\) for \(i \leq 2n+1\). Thus by examining the Serre spectral sequence for the fibre of \(X \to \Omega (X\wedge S^1)\) and using Hurewicz, we learn that the fibre \(F\) is \(2n\)-connected. □

Theorem 8.10 (Relative Hurewicz). Suppose that \(f:X \to Y\) is a map with fibre \(F\) and cofibre \(Y/X\). Suppose that \(f\) is \(n\)-connected, \(X\) is \(1\)-connected, and \(Y\) is connected. Then \(F\wedge S^1 \to Y/X\) is \(n+2\)-connected.

Proof. Observe first that \(Y\) is actually simply connected by assumption on \(f\). Thus the Serre spectral sequence for \(F \to X \to Y\) gives an exact sequence

\[H_{n+3}Y \xrightarrow {d_{n+3}} E_{n+3}^{0,n+2} \to H_{n+2}X \to H_{n+2}Y \xrightarrow {d_{n+2}} H_{n+1}F \to H_{n+1}X\to \cdots \]

From the description of the transgression, there is a comparison of exact sequences

The left hand map is surjective since every element of \(H_{n+3}Y\) is transgressive. The rest of the maps \(H_{i}(X/F)\to H_iY\) for \(i <n+3\) are isomorphisms by the \(5\)-lemma. Thus by comparing the long exact sequences on homology for \(X \to X/F \to \Sigma F\) and \(X \to Y \to Y/X\), we learn that the map \(F\wedge S^1 \to Y/X\) is \(n+2\)-connected. □

Corollary 8.11. Suppose that \(A\) is simply connected and the pair \((X,A)\) is \(n\)-connected for \(n\geq 1\). Then the Hurewicz map \(\pi _i(X,A) \to \tilde {H}_i(X,A)\) is an isomorphism for \(i \leq n+2\) and surjective for \(i = n+2\).

Proof. The relative homotopy groups are the homotopy groups of the homotopy fibre \(F\), and the Hurewicz map is the composite

\[\pi _i(F) \xrightarrow {\Sigma } \pi _{i+1}\Sigma F \rightarrow \pi _i(X/A) \to \tilde {H}_i(X/A)\]

, which is an isomorphism by a combination of Theorem 8.9 and the Theorem 8.10. □

Theorem 8.12. Let \(X\) be a pointed Kan complex and let \(\alpha _i,0 \leq i \leq n+1\) be pointed maps from \(\Delta ^n/\partial \Delta ^n \to X\). Then there is an \(n\)-simplex such that \(d_iw = \alpha _i\) iff \(\sum _0^{n+1} (-1)^{i}[\alpha _i] = 0\) in \(\pi _n(X)\).

Proof. It is easy for \(n =1\) by definition of multiplication on \(\pi _1\), so WLOG \(n \geq 2\). Consider the fibre sequence \(X_{\geq n} \to X \to X(n)\). \(F\) is \(n-1\)-connected, so we can assume by the homotopy extension property that each \(\alpha _i\) lies in the fibre. We can also assume that \(w\) lives in the fibre, since \(p(w)\) is nulhomotopy relative to the boundary, so we can choose a lift a nulhomotopy in \(X(n)\) to a homotopy in \(X\).

Thus we can assume \(X\) is \(n-1\)-connected. By the Hurewicz theorem, it follows that if we have such a \(w\), then \(\sum _0^{n+1} (-1)^{i}[\alpha _i] \) because it is \(dw\) in the Moore complex under the Hurewicz map.

Conversely, suppose that the formula is satisfied. we can form an extension in the diagram

By examining the homology class of \(d_0\theta \), we see it is the same as \(\alpha _0\), so by Hurewicz, it is homotopic to \(\alpha _0\). Thus by using the homotopy extension property, we can replace \(\theta \) with the desired simplex. □