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Ronnie Brown
Ronnie Brown
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In some sense, this is answered in the book Nonabelian Algebraic Topology (EMS Tracts in Math Vol 15, 2011). However this is not done for spaces but for filtered spaces $$X_*: X_0 \subseteq X_1 \subseteq \cdots \subseteq X_n \subseteq \cdots \subseteq X. $$ From this we construct a somewhat nonabelian analogue of a chain complex called a crossed complex $\Pi X_*$, using the fundamental groupoid $\pi_1(X_1,X_0)$ and relative homotopy groups    $$\pi_n(X_n,X_{n-1},x), x \in X_0$$ for $n \geqslant 2$, with with operations, and boundary maps. So this is nonabelian in dimensions $1$ and $2$. But since the relative homotopy groups are bydefined as homotopy classes of maps $(D^n,S^{n-1}, 0) \to (X_n,X_{n-1},x)$,$$(D^n,S^{n-1}, 0) \to (X_n,X_{n-1},x)$$ we are using "singular cells". (This construction on a filtered space was first considered with a different name by A L Blakers, Annals of Math., 1948, so it is quite classical.)

The problem is then how to compute $\Pi X_*$. Fortunately, there is a Seifert-van Kampen type theorem for $\Pi X_*$, allowing its determination or computation for filtered spaces built by gluing out of other filtered spaces. This includes the skeletal filtration of CW-complexes, giving a more powerful version of the cellular cochainschains of the universal cover of a CW-complex, and from which that operator cochainchain complex may be recovered.

The above theorem is not obtained directly but by using a cubical version, which we write $\rho X_*$, using homotopy classes rel vartices of maps $I^n_* \to X_*$, which we write $\rho X_*$, and which is shown to be a strict cubical higher homotopy groupoid. The compositions in $\rho X_*$ are intuitively natural constructions, but the proofs that they are well defined are non trivial. Once these compositions are obtained, the proof of a Seifert-van Kampen theorem for $\rho X_*$ follows the pattern of some proofs of the $1$-dimensional theorem. One also has to show that $\rho X_*$ is "equivalent" to $\Pi X_*$.

Other things are easier for $\rho$ thatthan for $\Pi$, particularly tensor products. For example a morphism $$\eta: \rho X_* \otimes \rho Y_* \to \rho (X_* \otimes Y_*)$$ is easily seen to be well defined by $[f] \otimes [g] \mapsto [f \otimes g]$, using $I^m_* \otimes I^n_* \cong I^{m+n}_*$, so the corresponding construction for $\Pi$ may be deduced.

There is a methodological argument for using filtered space, apart from the facts that they arise naturally, and that the theory works. To develop algebraic topology on a space %$X$, the space $X$ has to be given by some data. That data will have some kind of structure. It is not unreasonable to reflect that structure in terms of structure imposed on $X$. A filtration is one such example of structure.

Note that the results obtained by the theory do not require the setting up of simplicial homology, nor is simplicial approximation used at all, and most of the calculations obtained of relative homotopy groups in dimension $2$ are not obtainable by the use of abelian methods.

Some traditional results of homotopy theory, such as the simplicial Homotopy Addition Lemma, are also nicely expressed in the above terms.

The above results were all obtained with much collaboration from about 1965 by pursuing the question: if groupoids are useful in dimension $1$ homotopy theory, can they be useful in higher dimensional homotopy theory, using some form of multiple groupoids?

In some sense, this is answered in the book Nonabelian Algebraic Topology (EMS Tracts in Math Vol 15, 2011). However this is not done for spaces but for filtered spaces $$X_*: X_0 \subseteq X_1 \subseteq \cdots \subseteq X_n \subseteq \cdots \subseteq X. $$ From this we construct a somewhat nonabelian analogue of a chain complex called a crossed complex $\Pi X_*$, using the fundamental groupoid $\pi_1(X_1,X_0)$ and relative homotopy groups  $$\pi_n(X_n,X_{n-1},x), x \in X_0$$ for $n \geqslant 2$, with operations, and boundary maps. So this is nonabelian in dimensions $1$ and $2$. But since the relative homotopy groups are by homotopy classes of maps $(D^n,S^{n-1}, 0) \to (X_n,X_{n-1},x)$, we are using "singular cells". (This construction on a filtered space was first considered with a different name by A L Blakers, Annals of Math., 1948, so it is quite classical.)

The problem is then how to compute $\Pi X_*$. Fortunately, there is Seifert-van Kampen type theorem for $\Pi X_*$, allowing its determination or computation for filtered spaces built out of other filtered spaces. This includes the skeletal filtration of CW-complexes, giving a more powerful version of the cellular cochains of the universal cover of a CW-complex, and from which that operator cochain complex may be recovered.

The above theorem is not obtained directly but by using a cubical version, using homotopy classes rel vartices of maps $I^n_* \to X_*$, which we write $\rho X_*$, and which is shown to be a strict cubical higher homotopy groupoid. The compositions in $\rho X_*$ are intuitively natural constructions, but the proofs that they are well defined are non trivial. Once these compositions are obtained, the proof of a Seifert-van Kampen theorem for $\rho X_*$ follows the pattern of some proofs of the $1$-dimensional theorem. One also has to show that $\rho X_*$ is "equivalent" to $\Pi X_*$.

Other things are easier for $\rho$ that for $\Pi$, particularly tensor products. For example a morphism $$\eta: \rho X_* \otimes \rho Y_* \to \rho (X_* \otimes Y_*)$$ is easily seen to be well defined by $[f] \otimes [g] \mapsto [f \otimes g]$, using $I^m_* \otimes I^n_* \cong I^{m+n}_*$, so the corresponding construction for $\Pi$ may be deduced.

There is a methodological argument for using filtered space, apart from the facts that they arise naturally, and that the theory works. To develop algebraic topology on a space %$X$, the space $X$ has to be given by some data. That data will have some kind of structure. It is not unreasonable to reflect that structure in terms of structure imposed on $X$. A filtration is one such example of structure.

Note that the results obtained by the theory do not require the setting up of simplicial homology, nor is simplicial approximation used at all, and most of the calculations obtained of relative homotopy groups in dimension $2$ are not obtainable by the use of abelian methods.

Some traditional results of homotopy theory, such as the simplicial Homotopy Addition Lemma, are also nicely expressed in the above terms.

The above results were all obtained with much collaboration from about 1965 by pursuing the question: if groupoids are useful in dimension $1$ homotopy theory, can they be useful in higher dimensional homotopy theory, using some form of multiple groupoids?

In some sense, this is answered in the book Nonabelian Algebraic Topology (EMS Tracts in Math Vol 15, 2011). However this is not done for spaces but for filtered spaces $$X_*: X_0 \subseteq X_1 \subseteq \cdots \subseteq X_n \subseteq \cdots \subseteq X. $$ From this we construct a somewhat nonabelian analogue of a chain complex called a crossed complex $\Pi X_*$, using the fundamental groupoid $\pi_1(X_1,X_0)$ and relative homotopy groups  $$\pi_n(X_n,X_{n-1},x), x \in X_0$$ for $n \geqslant 2$, with operations, and boundary maps. So this is nonabelian in dimensions $1$ and $2$. But since the relative homotopy groups are defined as homotopy classes of maps $$(D^n,S^{n-1}, 0) \to (X_n,X_{n-1},x)$$ we are using "singular cells". (This construction on a filtered space was first considered with a different name by A L Blakers, Annals of Math., 1948, so it is quite classical.)

The problem is then how to compute $\Pi X_*$. Fortunately, there is a Seifert-van Kampen type theorem for $\Pi X_*$, allowing its determination or computation for filtered spaces built by gluing out of other filtered spaces. This includes the skeletal filtration of CW-complexes, giving a more powerful version of the cellular chains of the universal cover of a CW-complex, and from which that operator chain complex may be recovered.

The above theorem is not obtained directly but by using a cubical version, which we write $\rho X_*$, using homotopy classes rel vartices of maps $I^n_* \to X_*$, and which is shown to be a strict cubical higher homotopy groupoid. The compositions in $\rho X_*$ are intuitively natural constructions, but the proofs that they are well defined are non trivial. Once these compositions are obtained, the proof of a Seifert-van Kampen theorem for $\rho X_*$ follows the pattern of some proofs of the $1$-dimensional theorem. One also has to show that $\rho X_*$ is "equivalent" to $\Pi X_*$.

Other things are easier for $\rho$ than for $\Pi$, particularly tensor products. For example a morphism $$\eta: \rho X_* \otimes \rho Y_* \to \rho (X_* \otimes Y_*)$$ is easily seen to be well defined by $[f] \otimes [g] \mapsto [f \otimes g]$, using $I^m_* \otimes I^n_* \cong I^{m+n}_*$, so the corresponding construction for $\Pi$ may be deduced.

There is a methodological argument for using filtered space, apart from the facts that they arise naturally, and that the theory works. To develop algebraic topology on a space $X$, the space $X$ has to be given by some data. That data will have some kind of structure. It is not unreasonable to reflect that structure in terms of structure imposed on $X$. A filtration is one such example of structure.

Note that the results obtained by the theory do not require the setting up of simplicial homology, nor is simplicial approximation used at all, and most of the calculations obtained of relative homotopy groups in dimension $2$ are not obtainable by the use of abelian methods.

Some traditional results of homotopy theory, such as the simplicial Homotopy Addition Lemma, are also nicely expressed in the above terms.

The above results were all obtained with much collaboration from about 1965 by pursuing the question: if groupoids are useful in dimension $1$ homotopy theory, can they be useful in higher dimensional homotopy theory, using some form of multiple groupoids?

Source Link
Ronnie Brown
Ronnie Brown
  • 12.3k
  • 1
  • 63
  • 81

In some sense, this is answered in the book Nonabelian Algebraic Topology (EMS Tracts in Math Vol 15, 2011). However this is not done for spaces but for filtered spaces $$X_*: X_0 \subseteq X_1 \subseteq \cdots \subseteq X_n \subseteq \cdots \subseteq X. $$ From this we construct a somewhat nonabelian analogue of a chain complex called a crossed complex $\Pi X_*$, using the fundamental groupoid $\pi_1(X_1,X_0)$ and relative homotopy groups $$\pi_n(X_n,X_{n-1},x), x \in X_0$$ for $n \geqslant 2$, with operations, and boundary maps. So this is nonabelian in dimensions $1$ and $2$. But since the relative homotopy groups are by homotopy classes of maps $(D^n,S^{n-1}, 0) \to (X_n,X_{n-1},x)$, we are using "singular cells". (This construction on a filtered space was first considered with a different name by A L Blakers, Annals of Math., 1948, so it is quite classical.)

The problem is then how to compute $\Pi X_*$. Fortunately, there is Seifert-van Kampen type theorem for $\Pi X_*$, allowing its determination or computation for filtered spaces built out of other filtered spaces. This includes the skeletal filtration of CW-complexes, giving a more powerful version of the cellular cochains of the universal cover of a CW-complex, and from which that operator cochain complex may be recovered.

The above theorem is not obtained directly but by using a cubical version, using homotopy classes rel vartices of maps $I^n_* \to X_*$, which we write $\rho X_*$, and which is shown to be a strict cubical higher homotopy groupoid. The compositions in $\rho X_*$ are intuitively natural constructions, but the proofs that they are well defined are non trivial. Once these compositions are obtained, the proof of a Seifert-van Kampen theorem for $\rho X_*$ follows the pattern of some proofs of the $1$-dimensional theorem. One also has to show that $\rho X_*$ is "equivalent" to $\Pi X_*$.

Other things are easier for $\rho$ that for $\Pi$, particularly tensor products. For example a morphism $$\eta: \rho X_* \otimes \rho Y_* \to \rho (X_* \otimes Y_*)$$ is easily seen to be well defined by $[f] \otimes [g] \mapsto [f \otimes g]$, using $I^m_* \otimes I^n_* \cong I^{m+n}_*$, so the corresponding construction for $\Pi$ may be deduced.

There is a methodological argument for using filtered space, apart from the facts that they arise naturally, and that the theory works. To develop algebraic topology on a space %$X$, the space $X$ has to be given by some data. That data will have some kind of structure. It is not unreasonable to reflect that structure in terms of structure imposed on $X$. A filtration is one such example of structure.

Note that the results obtained by the theory do not require the setting up of simplicial homology, nor is simplicial approximation used at all, and most of the calculations obtained of relative homotopy groups in dimension $2$ are not obtainable by the use of abelian methods.

Some traditional results of homotopy theory, such as the simplicial Homotopy Addition Lemma, are also nicely expressed in the above terms.

The above results were all obtained with much collaboration from about 1965 by pursuing the question: if groupoids are useful in dimension $1$ homotopy theory, can they be useful in higher dimensional homotopy theory, using some form of multiple groupoids?